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A REVIEW OF ALGEBRA
BY
ROMEYN HENRY RIVENBURG, A.M.
HEAD OF THE DEPARTMENT OF MATHEMATICS
THE PEDDIE INSTITUTE, HIGHTSTOWN, N.J.
[Illustration]
AMERICAN BOOK COMPANY
NEW YORK CINCINNATI CHICAGO
COPYRIGHT, 1914,
BY ROMEYN H. RIVENBURG.
COPYRIGHT, 1914, IN GREAT BRITAIN.
A REVIEW OF ALGEBRA.
E. P. 6
PREFACE
In most high schools the course in Elementary Algebra is finished by the
end of the second year. By the senior year, most students have forgotten
many of the principles, and a thorough review is necessary in order to
prepare college candidates for the entrance examinations and for
effective work in the freshman year in college. Recognizing this need,
many schools are devoting at least two periods a week for part of the
senior year to a review of algebra.
For such a review the regular textbook is inadequate. From an
embarrassment of riches the teacher finds it laborious to select the
proper examples, while the student wastes time in searching for
scattered assignments. The object of this book is to conserve the time
and effort of both teacher and student, by providing a thorough and
effective review that can readily be completed, if need be, in two
periods a week for a half year.
Each student is expected to use his regular textbook in algebra for
reference, as he would use a dictionary,--to recall a definition, a
rule, or a process that he has forgotten. He should be encouraged to
_think_ his way out wherever possible, however, and to refer to the
textbook only when _forced_ to do so as a last resort.
The definitions given in the General Outline should be reviewed as
occasion arises for their use. The whole Outline can be profitably
employed for rapid class reviews, by covering the part of the Outline
that indicates the answer, the method, the example, or the formula, as
the case may be.
The whole scheme of the book is ordinarily to have a page of problems
represent a day's work. This, of course, does not apply to the Outlines
or the few pages of theory, which can be covered more rapidly. By this
plan, making only a part of the omissions indicated in the next
paragraph, the essentials of the algebra can be readily covered, if need
be, in from thirty to thirty-two lessons, thus leaving time for tests,
even if only eighteen weeks, of two periods each, are allotted to the
course.
If a brief course is desired, the Miscellaneous Examples (pp. 31 to 35,
50 to 52), many of the problems at the end of the book, and the College
Entrance Examinations may be omitted without marring the continuity or
the comprehensiveness of the review.
ROMEYN H. RIVENBURG.
CONTENTS
PAGES
OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA 7-13
ORDER OF OPERATIONS, EVALUATION, PARENTHESES 14
SPECIAL RULES OF MULTIPLICATION AND DIVISION 15
CASES IN FACTORING 16, 17
FACTORING 18
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE 19
FRACTIONS 20
COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS 21, 22
SIMULTANEOUS EQUATIONS AND INVOLUTION 23, 24
SQUARE ROOT 25
THEORY OF EXPONENTS 26-28
RADICALS 29, 30
MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS 31-35
QUADRATIC EQUATIONS 36, 37
THE THEORY OF QUADRATIC EQUATIONS 38-41
OUTLINE OF SIMULTANEOUS QUADRATICS 42, 43
SIMULTANEOUS QUADRATICS 44
RATIO AND PROPORTION 45, 46
ARITHMETICAL PROGRESSION 47
GEOMETRICAL PROGRESSION 48
THE BINOMIAL THEOREM 49
MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND 50-52
PROBLEMS--LINEAR EQUATIONS, SIMULTANEOUS EQUATIONS,
QUADRATIC EQUATIONS, SIMULTANEOUS QUADRATICS 53-57
COLLEGE ENTRANCE EXAMINATIONS 58-80
OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA
~Important Definitions~
Factors; coefficient; exponent; power; base; term; algebraic
sum; similar terms; degree; homogeneous expression; linear
equation; root of an equation; root of an expression; identity;
conditional equation; prime quantity; highest common factor (H.
C. F.); lowest common multiple (L. C. M.); involution;
evolution; imaginary number; real number; rational; similar
radicals; binomial surd; pure quadratic equation; affected
quadratic equation; equation in the quadratic form; simultaneous
linear equations; simultaneous quadratic equations;
discriminant; symmetrical expression; ratio; proportion; fourth
proportional; third proportional; mean proportional; arithmetic
progression; geometric progression; S [infinity]
~Special Rules for Multiplication and Division~
1. Square of the sum of two quantities.
(x + y)^2.
2. Square of the difference of two quantities.
(x - y)^2.
3. Product of the sum and difference of two quantities.
(s + t)(s - t).
4. Product of two binomials having a common term.
(x + r)(x + m).
5. Product of two binomials whose corresponding terms are similar.
(3x + 2t)(2x - 5t).
6. Square of a polynomial.
(m - n/3 + k)^2.
7. Sum of two cubes.
(x^3 + y^3)/(x + y) = x^2 - xy + y^2.
8. Difference of two cubes.
(x^3 - y^3)/(x - y) = x^2 + xy + y^2.
9. Sum or difference of two like powers.
(x^7 + y^7)/(x + y),
(x^5 - y^5)/(x - y),
(x^4 - y^4)/(x - y),
(x^4 - y^4)/(x + y).
~Cases in Factoring~
1. Common monomial factor.
mx + my - mz = m(x + y - z).
2. Trinomial that is a perfect square.
x^2 +- 2xy + y^2 = (x +- y)^2.
3. The difference of two squares.
(a) Two terms.
x^2 - y^2 = (x + y)(x - y).
(b) Four terms.
x^2 + 2xy + y^2 - m^2 = (x + y + m)(x + y - m).
(c) Six terms.
x^2 + 2xy + y^2 - p^2 - 2pq - q^2
= [(x + y) + (p + q)][(x + y) - (p + q)].
(d) Incomplete square.
x^4 + x^2 y^2 + y^4
= x^4 + 2x^2 y^2 + y^4 - x^2 y^2
= (x^2 + y^2 + xy)(x^2 + y^2 - xy).
4. Trinomial of the form x^2 + bx + c.
x^2 - 5x + 6 = (x - 2)(x - 3).
5. Trinomial of the form ax^2 + bx + c.
20x^2 + 7x - 6 = (4x + 3)(5x - 2).
6. Sum or difference of
two cubes. See "Special Rules," 7 and 8.
two like powers. See "Special Rules," 9.
7. Common polynomial factor. Grouping.
t^2 p + t^2 q - 2mp - 2mq
= t^2(p + q) - 2m(p + q) = (p + q)(t^2 - 2m).
8. Factor Theorem.
x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).
~H. C. F. and L. C. M.~
a^2 + 2a - 3 = (a + 3)(a - 1).
a^2 + 7a + 12 = (a + 3)(a + 4).
a^4 + 27a = a(a + 3)(a^2 - 3a + 9).
H. C. F. = a + 3.
L. C. M. = (a + 3)(a - 1)(a + 4)a(a^2 - 3a + 9).
~Fractions~
Reduction to lowest terms.
Reduction of a mixed number to an improper fraction.
Reduction of an improper fraction to a mixed number.
Addition and subtraction of fractions.
Multiplication and division of fractions.
Law of signs in division, changing signs of factors, etc.
Complex fractions.
~Simultaneous Equations~
Solved by
addition or subtraction.
substitution.
comparison.
Graphical representation.
~Involution~
Law of signs.
Binomial theorem laws.
Expansion of
monomials and fractions.
binomials.
trinomials.
~Evolution~
Law of signs.
Evolution of monomials and fractions.
Square root of algebraic expressions.
Square root of arithmetical numbers.
Optional
Cube root of algebraic expressions.
Cube root of arithmetical numbers.
~Theory of Exponents~
Proofs: a^m x a^n = a^(m + n);
(a^m)/(a^n) = a^(m - n);
(a^m)^n = a^(mn);
[a^(mn)]^(1/n) = a^m;
(a/b)^n = (a^n)/(b^n);
(abc)^n = a^n b^n c^n.
Meaning of
fractional exponent.
zero exponent.
negative exponent.
Four rules
To multiply quantities having the same base, add exponents.
To divide quantities having the same base, subtract
exponents.
To raise to a power, multiply exponents.
To extract a root, divide the exponent of the power by the
index of the root.
~Radicals~
Radical in its simplest form.
Transformation of radicals
Fraction under the radical sign.
Reduction to an entire surd.
Changing to surds of different order.
Reduction to simplest form.
Addition and subtraction of radicals.
Multiplication and division of radicals
a^(1/n) . b^(1/n) = [ab]^(1/n).
([ab]^(1/n))/(a^(1/n)) = b^(1/n).
Rationalization
Monomial denominator.
Binomial denominator.
Trinomial denominator.
Square root of a binomial surd.
Radical equations. _Always_ check results to avoid extraneous roots.
~Quadratic Equations~
Pure. x^2 = a.
Affected. ax^2 + bx + c = 0.
Methods of solving
Completing the square.
Formula. Developed from ax^2 + bx + c = 0.
Factoring.
Equations in the quadratic form.
Properties of quadratics
r_1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a).
r_2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a).
Then
r_1 + r_2 = -b/a.
r_1 . r_2 = c/a.
Discriminant, b^2 - 4ac, and its discussion.
Nature or character of the roots.
~Simultaneous Quadratics~
CASE I.
One equation linear.
The other quadratic.
3x - y = 12,
x^2 - y^2 = 16.
CASE II.
Both equations homogeneous and of the second degree.
x^2 - xy + y^2 = 21,
y^2 - 2xy = -15.
CASE III.
Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3,
x^3 - y^3, x - y, x^2 +- xy + y^2, etc., given.
x^2 + y^2 = 41,
x + y = 9.
CASE IV.
Both equations symmetrical or symmetrical except for sign.
Usually one equation of high degree, the other of the first
degree.
x^5 + y^5 = 242,
x + y = 2.
CASE V. Special Devices
I. Solve for a compound unknown, like xy, x + y, (1)/(xy),
etc., first.
x^2y^2 + xy = 6,
x + 2y = -5.
II. Divide the equations, member by member.
x^4 - y^4 = 20,
x^2 - y^2 = 5.
III. Eliminate the quadratic terms.
4x + 3y = 2xy,
7x - 5y = 5xy.
~Ratio and Proportion~
Proportionals
mean,
third,
fourth.
Theorems
1. Product of means equals product of extremes.
2. If the product of two numbers equals the product of two
other numbers, either pair, etc.
3. Alternation.
4. Inversion.
5. Composition.
6. Division.
7. Composition and division.
8. In a series of equal ratios, the sum of the antecedents
is to the sum of the consequents as any antecedent, etc.
Special method of proving four quantities in proportion. Let
a/b = x, a = bx, etc.
~Progressions~
Development of formulas.
{ l = ar^(n - 1).
{ l = a + (n - 1)d. { S = (ar^n - a)/(r - 1).
{ S = (n/2)(a + l). { S = (rl - a)/(r - 1).
{ S = (n/2)[2a + (n - 1)d]. { S[infinity] = (a)/(1 - r).
Insertion of means
Arithmetical.
Geometrical.
~Binomial Theorem~
Review of binomial theorem laws. See Involution.
Expansion of (a + b)^n.
Finding any term by
key number method.
r^(th) or (r + 1)^(th) term method.
A REVIEW OF ALGEBRA
ORDER OF OPERATIONS, EVALUATION, PARENTHESES
Order of operations:
First of all, raising to a power and extracting a root.
Next, multiplication and division.
Last of all, addition and subtraction.
Find the value of:
1. 5 . 2^2 - 25^(1/2) / 5 + 2^2 . 8 / 4 - 2.
2. (3 x 6 / 9)/2 - 2[100^(1/2)] / 5 + 4 . 2^3 - (14 . 2)/28.
3. 9 . 2 / 6 + 3 - 2 . 4^2 / 8^(1/3) - 4 + (3 . 2^2)/6.
Evaluate:
4. (a^4 - a^3 + b^3)/([a^2 b^2]^(1/2)) + (c[a^(1/2)] + a^3bc)/(abc),
if a = 1, b = 2, c = 3.
5. t^(1/3) + [tm]^(1/3) + m^(1/3), if t = 8, m = 27.
6. (2[3 + 2d + a]^(1/2))/(3[a + b - cx - c]^(1/2))
+ ((3c - d)x)/(7ad - [abc]^(1/2)), if
a = 5, b = 3, c = -1, d = -2, x = 0.
7. a - {5b - [a - (3c - 3b) + 2c - 3(a - 2b - c)]},
if a = -3, b = 4, c = -5. (_Yale._)
Simplify:
8. m - [2m - {3r - (4r - 2m)}].
9. 2a - [5d + {3c - (a + [2d - 3a + 4c])}].
10. 3c^2 + c(2a - [6c - {3a + c - 4a}]).
SPECIAL RULES OF MULTIPLICATION AND DIVISION
Give results by inspection:
1. (g + 1/2 k)^2.
2. (s - (2m)/3)^2.
3. (2v + 3w)(2v - 3w).
4. (x + 3ts)(x - 7ts).
5. (2l + 3g)(4l - 11g).
6. (a - (2b)/3 + c - d)^2.
7. (x^3 + 8m^3)/(x + 2m).
8. (y^3 - 27k^(3m))/(y - 3k^m).
9. (c^5 - d^5)/(c - d).
10. (e^5 + d^5)/(e + d).
11. (x^4 - y^4)/(x - y).
12. (x^4 - y^4)/(x + y).
13. (a - .03)(a - .0007).
14. (g^n - 1/2)(g^n + 3/4).
15. (t^7 - v^(7/2))/(t - v^(1/2)).
16. (k^32 + 1)(k^16 +1)(k^8 + 1)(k^4 +1)(k^2 +1)(k + 1)(k - 1).
17. [(a + b) + (c + d)][(a + b) - (c + d)].
18. (p - q + r - s)(p - q - r + s).
19. (3m - n - l + 2r)(3m + n - l - 2r).
20. (x + 5)(x - 2)(x - 5)(x + 2).
21. (a^2 + b^2 - c - 2d + 3e)^2.
22. (s + t - 2v/5 + 3w/6 + z^2)^2.
23. (x^5 + 32)/(x + 2).
~References:~ The chapter on Special Rules of Multiplication and
Division in any algebra.
Special Rules of Multiplication and Division in the
Outline in the front of the book.
CASES IN FACTORING
The number of terms in an expression usually gives the clue to the
possible cases under which it may come. By applying the _test_ for each
and eliminating the _possible_ cases one by one, the right case is
readily found. Hence, the number of terms in the expression and a ready
and accurate knowledge of the Cases in Factoring are the real keys to
success in this vitally important part of algebra.
CASE I. A common monomial factor. Applies to any number of terms.
5cx - 5ct + 5cv - 15 c^2 m + 25 c^3 m^2
= 5c(x - t + v - 3cm + 5c^2 m^2).
CASE II. A trinomial that is a perfect square. Three terms.
x^2 +- 2xm + m^2 = (x +- m)^2.
CASE III. The difference of two squares.
_A._ Two terms. x^2 - y^2 = (x + y)(x - y).
_B._ Four terms.
x^2 + 2xy + y^2 - m^2 = (x^2 + 2xy + y^2) - m^2
= (x + y + m)(x + y - m)
_C._ Six terms. x^2 - 2xy + y^2 - m^2 - 2mn - n^2
= (x^2 - 2xy + y^2) - (m^2 + 2mn + n^2)
= (x - y)^2 - (m + n)^2
= [(x - y) + (m + n)][(x - y) - (m + n)].
_D._ An incomplete square. Three terms, and
4th powers or multiples of 4.
c^4 + c^2 d^2 + d^4 = c^4 + 2c^2 d^2 + d^4 - c^2 d^2
= (c^2 + d^2)^2 - c^2 d^2
= (c^2 + d^2 + cd)(c^2 + d^2 - cd).
CASE IV. A trinomial of the form x^2 + bx + c. Three terms.
x^2 + x - 30 = (x + 6)(x - 5).
CASE V. A trinomial of the form ax^2 + bx + c. Three terms.
20x^2 + 7x - 6 = (4x + 3)(5x - 2).
CASE VI.
_A._ The sum or difference of two cubes. Two terms.
x^3 + y^3 = (x + y)(x^2 - xy + y^2);
x^3 - y^3 = (x - y)(x^2 + xy + y^2).
_B._ The sum or difference of two like powers. Two terms.
x^4 - y^4 = (x - y)(x^3 + x^2y + xy^2 + y^3);
x^5 + y^5 = (x + y)(x^4 - x^3y + x^2 y^2 - xy^3 + y^4).
CASE VII. A common polynomial factor. Any _composite_ number of terms.
t^2 p + t^2 q - t^2 r - g^2 p - g^2 q + g^2 r
= t^2 (p + q - r) - g^2 (p + q -r)
= (p + q - r)(t^2 - g^2)
= (p + q - r)(t + g)(t - g).
CASE VIII. The Factor Theorem. Any number of terms.
x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).
FACTORING
Review the _Cases in Factoring_ (see Outline on preceding pages) and
write out the prime factors of the following:
1. 8a^(13) + am^(12).
2. x^7 + y^7.
3. 4x^2 + 11x - 3.
4. m^2 + n^2 - (1 + 2mn).
5. -x^2 + 2x - 1 + x^4.
6. x^(16) - y^(16). (Five factors.)
7. (x + 1)^2 - 5x - 29.
8. x^4 + x^2 y^2 + y^4.
9. x^4 - 11x^2 + 1.
10. x^(2m) + 2 + 1/(x^(2m)).
11. x^(6m) + 13x^(3m) + 12.
12. 4a^2 b^2 - (a^2 + b^2 - c^2)^2.
13. (x^2 - x - 6)(x^2 - x - 20).
14. a^4 - 8a - a^3 + 8.
15. p^3 + 7p^2 + 14p + 8.
16. 18a^2 b + 60ab^2 + 50b^3.
17. x^3 - 7x + 6.
18. 24c^2 d^2 - 47cd - 75.
19. (a^2 - b^2)^2 - (a^2 - ab)^2.
20. a^2 x^3 - (8a^2)/(y^3) - x^3 + 8/(y^3).
21. gt - gk + gl^2 + xt - xk + xl^2.
22. (m - n)(2a^2 - 2ab) + (n - m)(2ab - 2b^2).
23. a^2 - x^2 - y^2 + b^2 + 2ab + 2xy.
24. (2c^2 + 3d^2)a + (2a^2 + 3c^2)d.
25. (n(n - 1))/(1 . 2) a^(n - 2) b^2
+ (n(n - 1)(n - 2))/(1 . 2 . 3) a^(n - 3) b^3.
26. (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_M. I. T._)
27. (27y^3)^2 - 2(27y^3)(8b^3) + (8b^3)^2. (_Princeton._)
28. (a^3 + 8b^3)(a + b) - 6ab(a^2 - 2ab + 4b^2). (_M. I. T._)
Solve by factoring:
29. x^3 = x.
30. z^2 - 4z - 45 = 0.
31. x^3 - x^2 = 4x - 4.
~Reference:~ The chapter on Factoring in any algebra.
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
Define H. C. F. and L. C. M.
Find by factoring the H. C. F. and L. C. M.:
1. 3x^2 - 3x,
12x^2 (x^2 - 1),
18x^3 (x^3 - 1).
2. (x^2 - 1)(x^2 + 5x + 6),
(x^2 + 3x)(x^2 - x - 6). (_Harvard._)
3. x^2 - y^2,
x^2 + y^2,
x^3 + y^3,
x^6 + y^6,
x^6 - y^6. (_College Entrance Board._)
4. x^3 + x^2 - 2,
x^3 + 2x^2 - 3. (_Cornell._)
5. x^5 - 2x^4 + x^2,
2x^4 - 4x^3 - 4x + 6. (_Yale._)
6. x^2 + a^2 - b^2 + 2ax,
x^2 - a^2 + b^2 + 2bx,
x^2 - a^2 - b^2 - 2ab. (_Harvard._)
7. 2x^2 - x - 15,
3x^2 - 11x + 6,
2x^3 - x^2 - 13x - 6. (_College Entrance Board._)
8. (tv - v^2)^3,
v^3 - t^2v,
t^3 - v^3,
v^2 - 2vt + t^2.
Pick out the H. C. F. and the L. C. M. of the following:
9. 8(x^2 + y)^(32) (t^2 + z)^(19) (m - n^3)^(14),
12(x^2 + y)^(23) (t^2 + z)^(41) (m - n^3)^(17),
18(m - n^3)^(11) (x^2 + y)^(39) (t^2 + z)^(37).
10. 17ax^3 (y + z)^(10) (y - x)^(19) (x + z)^(27),
34a^2 x^4 (y + z)^(11) (y - x)^(21) (x + z)^(13),
51a^3 x^5 (y + z)^4 (x + z)^(32) (y - x)^(29).
~Reference:~ The chapter on H. C. F. and L. C. M. in any algebra.
FRACTIONS
Define: fraction, terms of a fraction, reciprocal of a number.
Look up _the law of signs_ as it applies to fractions. Except for this,
fractions in algebra are treated exactly the same as they are in
arithmetic.
1. Reduce to lowest terms:
(_a_) 32/24;
(_b_) (a^6 - x^6)/(a^4 - x^4);
(_c_) [(a + b)^2 - (c + d)^2]/[(a + c)^2 - (b + d)^2]. (_M. I. T._)
2. Reduce to a mixed expression:
(_a_) 756/11;
(_b_) (a^3 + b^3)/(a - b).
3. Reduce to an improper fraction:
(_a_) 45-1/8;
(_b_) 9-11/12 qt.;
(_c_) a^2 - ab + b^2 - (b^3)/(a + b).
Add:
4. 5/18 + 7/9 + 11/16 + 5/8.
5. 5/(1 + 2x) - (3x)/(1 - 2x) + (4 - 13x)/(4x^2 - 1).
6. 1/[x(x - a)(x - b)] + 1/[a(a - x)(a - b)] + 1/[b(b - x)(b - a)].
Multiply:
7. 72/121 x 55/56 x 77/90.
8. (b - y)/(a^3 + y^3) x (ca + cy)/(b^2 + by)
x (b^6 + y^6)/(b^2 + y^2) x b/c.
Divide:
9. (12/25) / (6/50).
10. [1 - (ab)/(a^2 - ab + b^2)] [1 - (ab)/(a^2 + 2ab + b^2)]
/ (a^3 - b^3)/(a^3 + b^3). (_Yale._)
11. [(x^4 - y^4)/(x^2 - y^2) / (x + y)/(x^2 - xy)]
/ [(x^2 + y^2)/(x - y) / (x + y)/(xy - y^2)]. (_Sheffield._)
Simplify:
12. [(4y)/x - (15y^2)/(x^2) + 4] / [4 - (16y)/x + (15y^2)/(x^2)]
x [3 - (4x + 20y)/(2x + 5y)].
~Reference:~ The chapter on Fractions in any algebra.
COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS
Define a complex fraction.
Simplify:
1. (3/7 + 4/5)/(2 - 3/7 . 4/5).
2. (2 - 3/2 + 2/3)/(5 - 2/3 + 3/2).
3. 2 - 2/(1 - 1/[1 - 1/(1 + 1/2)]).
4. a/(b^2) - a/[b^2 + (cb)/(a - c/b)]. (_Harvard._)
5. If m = 1/(a + 1), n = 2/(a + 2), p = 3/(a + 3), what is the value
of m/(1 - m) + n/(1 - n) + p/(1 - p)? (_Univ. of Penn._)
6. Simplify the expression
{x + y - 1/[x + y - xy/(x + y)]}(x^3 - y^3)/(x^2 - y^2).
(_Cornell._)
7. Simplify [1 - (2xy)/((x + y)^2)]/[1 + (2xy)/((x - y)^2)]
/ {(1 - y/x)/(1 + y/x)}^2.
8. Solve (7y + 9)/4 - [y - (2y - 1)/9] = 7.
9. Solve 2-1/3 - (2/5)(x^2 + 3) = (10x)/3 + 1 - (2x^2)/5.
10. How much water must be added to 80 pounds of a 5 per cent salt
solution to obtain a 4 per cent solution? (_Yale._)
~Reference:~ See Complex Fractions, and the first part of the chapter on
Fractional Equations in any algebra.
FRACTIONAL EQUATIONS
1. Solve for each letter in turn 1/b = 1/p + 1/q.
2. Solve and check:
(5x + 2)/3 - (3 - (3x - 1)/2) = (3x + 19)/2 - ((x + 1)/6 + 3).
3. Solve and check:
(1/2)(x - a/3) - (1/3)(x - a/4) + (1/4)(x - a/5) = 0.
4. Solve (after looking up the special _short_ method):
(3x - 1)/30 + (4x - 7)/15
= x/4 - (2x - 3)/(12x - 11) + (7x - 15)/60.
5. Solve by the special _short_ method:
1/(x - 2) - 1/(x - 3) = 1/(x - 4) - 1/(x - 5).
6. At what time between 8 and 9 o'clock are the hands of a watch
(_a_) opposite each other? (_b_) at right angles? (_c_) together?
Work out (_a_) and state the equations for (_b_) and (_c_).
7. The formula for converting a temperature of F degrees Fahrenheit
into its equivalent temperature of C degrees Centigrade is
C = (5/9)(F - 32). Express F in terms of C, and compute F for the
values C = 30 and C = 28. (_College Entrance Exam. Board._)
8. What is the price of eggs when 2 less for 24 cents raises the price
2 cents a dozen? (_Yale._)
9. Solve 2/(x - 2) + 2/(4 - x^2) = 5/(x + 2).
~Reference:~ The Chapter on Fractional Equations in any algebra. Note
particularly the special _short_ methods, usually given
about the middle of the chapter.
SIMULTANEOUS EQUATIONS
NOTE. Up to this point each topic presented has reviewed to some
extent the preceding topics. For example, factoring reviews the
special rules of multiplication and division; H. C. F. and L. C. M.
review factoring; addition and subtraction of fractions and
fractional equations review H. C. F. and L. C. M., etc. From this
point on, however, the interdependence is not so marked, and
miscellaneous examples illustrating the work already covered will
be given very frequently in order to keep the whole subject fresh
in mind.
1. Solve by three methods--addition and subtraction, substitution,
and comparison: { 5x + y = 11,
{ 3x + 2y = 1.
Solve and check:
2. { 12R_1 - 11R_2 = b + 12c,
{ R_1 + R_2 = 2b + c.
3. { (r - s)/2 = 25/6 - (r + s)/3,
{ (r + s - 9)/2 - (s - r - 6)/3 = 0.
4. One half of A's marbles exceeds one half of B's and C's together by
2; twice B's marbles falls short of A's and C's together by 16; if
C had four more marbles, he would have one fourth as many as A and
B together. How many has each? (_College Entrance Board._)
5. The sides of a triangle are a, b, c. Calculate the radii of the
three circles having the vertices as centers, each being tangent
externally to the other two. (_Harvard._)
6. Solve { 2x + 3y = 7, x - y = 1 } graphically; then solve
algebraically and compare results. (Use cooerdinate or squared
paper.)
Factor:
7. x^4 + 4.
8. 2d^(10) - 1024d.
9. 2(x^3 - 1) - 7(x^2 - 1).
~References:~ The chapters on Simultaneous Equations and Graphs in any
algebra.
SIMULTANEOUS EQUATIONS AND INVOLUTION
1. Solve (3/4)x - (5/3)y = 11-1/2,
(5/8)x - (3/2)y = 10-1/4.
Look up the method of solving when the unknowns are in the
denominator. Should you clear of fractions?
2. Solve 1/x - 1/y - 1/z = 1/a,
1/y - 1/z - 1/x = 1/b,
1/z - 1/x - 1/y = 1/c.
3. Solve graphically and algebraically
2x - y = 4,
2x + 3y = 12.
4. Solve graphically and algebraically
3x + 7y = 5,
8x + 3y = -18.
Review:
5. The squares of the numbers from 1 to 25.
6. The cubes of the numbers from 1 to 12.
7. The fourth powers of the numbers from 1 to 5.
8. The fifth powers of the numbers from 1 to 3.
9. The binomial theorem laws. (See Involution.)
Expand: (Indicate first, then reduce.)
10. (b + y)^7.
11. [(2a)/3 - 1]^5.
12. (x^2 + 2a)^5.
13. (x - y + 2z)^3.
14. A train lost one sixth of its passengers at the first stop, 25 at
the second stop, 20% of the remainder at the third stop, three
quarters of the remainder at the fourth stop; 25 remain. What was
the original number? (_M. I. T._)
~References:~ The chapter on Involution in any algebra. Also the
references on the preceding page.
SQUARE ROOT
Find the square root of:
1. 1 + 16m^6 - 40m^4 + 10m - 8m^3 + 25m^2.
2. (a^2)/(x^2) + (6a)/x + 11 + (6x)/a + (x^2)/(a^2).
3. Find the square root to three terms of x^2 + 5.
4. Find the square root of 337,561.
5. Find the square root of 1823.29.
6. Find to four decimal places the square root of 1.672.
(_Princeton._)
7. Add 2/[(x - 1)^3] + 1/[(1 - x)^2] - 2/(1 - x) - 1/x.
8. Find the value of:
(64^(1/3) . 12)/24 / 2 x 3 - (2 . 7^2)/(14) / 7
x 1 + (1^(1/3) . 1^7)/(1 . 1^2) - 4 . 0.
9. Simplify [(x + y)^5 + (x - y)^5][(x + y)^5 - (x - y)^5].
10. Solve by the short method:
5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0.
11. It takes 3/4 of a second for a ball to go from the pitcher to the
catcher, and 1/2 of a second for the catcher to handle it and get
off a throw to second base. It is 90 feet from first base to
second, and 130 feet from the catcher's position to second. A
runner stealing second has a start of 13 feet when the ball leaves
the pitcher's hand, and beats the throw to the base by 1/8 of a
second. The next time he tries it, he gets a start of only 3-1/2
feet, and is caught by 6 feet. What is his rate of running, and
the velocity of the catcher's throw? (_Cornell._)
~Reference:~ The chapter on Square Root in any algebra.
THEORY OF EXPONENTS
Review the proofs, for positive integral exponents, of:
I. a^m x a^n = a^(m + n).
II. (a^m)/(a^n) = a^(m - n).
III. (a^m)^n = a^(mn).
IV. [a^(mn)]^(1/n) = a^m.
V. [a/b]^n = (a^n)/(b^n).
VI. (abc)^n = a^n b^n c^n.
~To find the meaning of a fractional exponent.~
Assume that Law I holds for _all_ exponents.
If so, a^(2/3) . a^(2/3) . a^(2/3) = a^(6/3) = a^2.
Hence, a^(2/3) is _one of the three equal factors_ (hence the cube root)
of a^2.
Therefore a^(2/3) = [a^2]^(1/3).
In the same way,
a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) = a^(20/5) = a^4.
Hence, a^(4/5) is _one of the five equal factors_ (hence the fifth root)
of a^4.
Therefore a^(4/5) = [a^4]^(1/5).
In the same way, in general, a^(p/q) = [a^p]^(1/q).
Hence, _the numerator of a fractional exponent indicates the power, the
denominator indicates the root_.
~To find the meaning of a zero exponent.~
Assume that Law II holds for _all_ exponents.
If so, (a^m)/(a^m) = a^(m - m) = a^0. But by division, (a^m)/(a^m) = 1.
Therefore a^0 = 1. Axiom I.
~To find the meaning of a negative exponent.~
Assume that Law I holds for _all_ exponents.
If so, a^m x a^(-m) = a^(m - m) = a^0 = 1.
Hence, a^m x a^(-m) = 1.
Therefore a^(-m) = 1/(a^m).
Rules:
_To multiply quantities having the same base, add exponents._
_To divide quantities having the same base, subtract exponents._
_To raise a quantity to a power, multiply exponents._
_To extract a root, divide the exponent of the power by the
index of the root._
1. Find the value of 3^2 - 5 x 4^0 + 8^(-2/3) + 1^(2/5).
2. Find the value of 8^(-2/3) + 9^(3/2) - 2^(-2) + 1^(-2/5) - 7^0.
Give the value of each of the following:
3. (3^0)/5, 3/(5^0), (3^0)/(5^0),
3^0 x 5, 3 x 5^0, 3^0 x 5^0, 3^0 + 5^0, 3^0 - 5^0.
4. Express 7^0 as some power of 7 divided by itself.
Simplify:
5. 16^(1/3) . 2^(1/2) . 32^(5/6). (Change to the same base first.)
6. [2/(8^(-3))]^(1/5).
7. [(x^n)^(n + 2)]/[(x^(n + 1))(x^(n - 1))].
8. (x + 3x^(2/3) - 2x^(1/3))(3 - 2x^(-1/3) + 4x^(-2/3)).
9. [(a^2b)/(c^2d)]^(1/2) x [(c^3d)/(ab^3)]^(1/3)
x [(a^(1/3)c)/(b^(1/4)d^(5/12))]^2.
10. [(a^(-4))/(b^(-2)c)]^(-3/4)
x [(a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2).
11. [([a^2]^(1/3))/([b^(-1)]^(1/4)) . ([c^(-3)]^(1/2))/(a^(1/3))
. (b^(-1/4)a^(1/3))/(c^(-1))]^(-6).
~Reference:~ The chapter on Theory of Exponents in any algebra.
Solve for x:
1. x^(2/3) = 4.
2. x^(-3/4) = 8.
Factor:
3. x^(2/3) - 9.
4. x^(3/5) + 27.
5. x^(2a) - y^(-6).
6. a^(1/3) x^(1/2) - 3a^(1/3) + 5x^(1/2) - 15.
7. Find the H. C. F. and L. C. M. of
a^2 + a^(3/2) b^(1/2) + a^(1/2) b^(3/2) - b^2,
a^2 - a^(3/2) b^(1/2) - a^(1/2) b^(3/2) - b^2.
8. Simplify the product of:
(ayx^(-1))^(1/2), (bxy^(-2))^(1/3),
and (y^2a^(-2)b^(-2))^(1/4). (_Princeton._)
9. Find the square root of:
25a^(4/3)b^(-3) - 10a^(2/3)b^(-3/2) - 49
+ 10a^(-2/3)b^(3/2) + 25a^(-4/3)b^3.
10. Simplify [(2^(n + 2))/(4^(-n)) / (8^n)/(2^3)]^(1/5).
11. Find the value of
(7 . 13^0 / 7)/(21^0) + 3^0 x (4^0 . 7^0)/[(7a + b)^0] + 8^(-2/3).
12. Express as a power of 2: 8^3; 4^5; 4^3 . 8^(2/3) . 16^(3/4).
13. Simplify
{[(x^(a + 1))/(x^(1 - a))]^a
/ [(x^a)/(x^(1 - a))]^(a - 1)}^(1/(3a - 1)).
14. Simplify
[(x^(5/2) y^(4/3))/(z^(-5/4)) . (z^4)/(x^(-3) y^(-5/3))
/ (y^(-2) z^(1/4))/(x^(-1/2))]^(1/5).
15. Expand (a^(1/2) + b^(1/3))^4, writing the result with fractional
exponents.
~Reference:~ The chapter on Theory of Exponents in any algebra.
RADICALS
1. Review all definitions in Radicals, also the methods of
transforming and simplifying radicals. When is _a radical in its
simplest form_?
2. Simplify (to simplest form):
[2/3]^(1/2);
[1/11]^(1/2);
[3/5]^(1/3);
3[5/6]^(1/2);
(2a/b)[(8b^2)/(27a)]^(1/2);
[5/(x^n)]^(1/2n);
(a + b)^2 [(-a^4)/((a + b)^5)]^(1/3);
27^(1/2);
[54]^(1/3);
-5[125^(1/2)].
3. Reduce to entire surds:
2[3^(1/2)];
2[3^(1/4)];
6[2^(1/3)];
a[[b^2]^(1/n)];
-3[2^(1/3)];
3a[[(a + 2)/(6a^2)]^(1/3)];
(a + 2y)[(a - 2y)/(a + 2y)]^(1/2).
4. Reduce to radicals of lower order (or simplify indices):
[a^2]^(1/4);
[a^3]^(1/6);
[27a^3]^(1/6);
[81 a^4 x^8]^(1/12);
[9x^2 y^4 z^10]^(1/2n).
5. Reduce to radicals of the same degree (order, or index):
7^(1/2) and [11]^(1/3);
5^(1/3) and 3^(1/4);
7^(1/6) and 3^(1/2);
[x^m]^(1/n) and [x^n]^(1/m);
[c^y]^(1/x), [c^z]^(1/y), and [c^x]^(1/z).
6. Which is greater, 3^(1/2) or 4^(1/3)? [23]^(1/3) or 2[2^(1/2)]?
7. Which is greatest, 3^(1/2), 5^(1/3), or 7^(1/4)? Give work and
arrange in descending order of magnitude.
Collect:
8. 128^(1/2) - 2[50^(1/2)] + 72^(1/2) - 18^(1/2).
9. 2[5/3]^(1/2) + (1/6)60^(1/2) + 15^(1/2) + [3/5]^(1/2).
10. [(m - n)^2a]^(1/2) + [(m + n)^2a]^(1/2) - [am^2]^(1/2)
+ [a(n - m)^2]^(1/2) - a^(1/2).
11. A and B each shoot thirty arrows at a target. B makes twice as
many hits as A, and A makes three times as many misses as B.
Find the number of hits and misses of each. (_Univ. of Cal._)
~Reference:~ The chapter on Radicals in any algebra (first part of the
chapter).
The most important principle in Radicals is the following:
(ab)^(1/n) = a^(1/n) b^(1/n).
Hence [ab]^(1/n) = a^(1/n) . b^(1/n).
Or, a^(1/n) . b^(1/n) = [ab]^(1/n).
From this also ([ab]^(1/n))/(a^(1/n)) = b^(1/n).
Multiply:
1. 2[4^(1/3)] by 3[6^(1/3)].
2. 2^(1/2) by 3^(1/3).
3. 2^(1/4) by 4^(1/6).
4. [a + x^(1/2)]^(1/2) by [a - x^(1/2)]^(1/2).
5. 2^(1/2) + 3^(1/2) - 5^(1/2) by 2^(1/2) - 3^(1/2) + 5^(1/2).
6. -p/2 + ([p^2 - 4q]^(1/2))/2 by -p/2 - ([p^2 - 4q]^(1/2))/2.
Divide:
7. 27^(1/2) by 3^(1/2).
8. 4[18^(1/2)] by 5[32^(1/2)].
9. 3[12]^(1/3) by 6^(1/2).
10. 3^(1/2) by 3^(1/4).
11. 6[105^(1/2)] + 18[40^(1/2)] - 45[12^(1/2)] by 3[15^(1/2)].
(_Short division._)
12. 10[18]^(1/3) - 4[60]^(1/3) + 5[100]^(1/3) by 3[30]^(1/3).
Rationalize the denominator:
13. 2/(3^(1/2));
7/(7^(1/2));
5/(2[5^(1/2)]);
3/([a^2]^(1/5));
4/([a^3]^(1/7)).
14. 2/(2^(1/2)) + 3^(1/2));
(a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2));
3/(3 - 3^(1/2)).
15. [3^(1/2) + 2^(1/2)]/[6^(1/2) + 3^(1/2) - 2^(1/2)].
Review the method of finding the square root of a binomial surd. (By
inspection preferably.) Then find square root of:
16. 5 + 2[6^(1/2)].
17. 17 - 12[2^(1/2)].
18. 7 - 33^(1/2).
~Reference:~ The chapter on Radicals in any algebra, beginning at
Addition and Subtraction of Radicals.
MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS
Results by inspection, examples 1-10.
Divide:
1. (x^(5/17) + y^(5/17))/(x^(1/17) + y^(1/17)).
2. (x - y)/(x^(1/3) - y^(1/3)).
3. (m^2 + n^2)/(m^(2/3) + n^(2/3)).
4. (x - y^2)/(x^(1/3) - [y^2]^(1/3)).
Multiply:
5. [a^(-3/4) + 2/(m^(1/2))]^2.
6. (K^(-2/7) - g^(-11/25))^2.
7. (r^(2s) + l^(-3m))(r^(2s) - l^(-3m)).
8. [a^(-2) + b^(-3) - 1/(c^2)]^2.
9. (3K^x + 4t^(-3))(3K^x - 7t^(-3)).
10. (2y^(2/7) - 40K^3)(3y^(2/7) + 55K^3).
Factor:
11. x^(2/3) - 64.
12. y^(3/5) + 27.
13. b^(3/2) - 8m^(-1).
14. 3p - 8p^(1/2) - 35.
Factor, using radicals instead of exponents:
15. 60 - 7[3b^(1/2)] - 6b.
16. 15m - 2[[mn]^(1/2)] - 24n.
17. a - b (factor as difference of two squares).
18. a - b (factor as difference of two cubes).
19. a - b (factor as difference of two fourth powers).
20. Find the H. C. F. and L. C. M. of
x^2 + xy^(1/2) - 2y,
2x^2 + 5xy^(1/2) + 2y,
2x^2 - xy^(1/2) - y.
21. Solve (short method)
(x - 7)/(x - 8) - (x - 8)/(x - 9)
= (x - 4)/(x - 5) - (x - 5)/(x - 6).
22. Simplify
(ab/c + bc/a + ca/b)/(a/bc + b/ca + c/ab)
x [((a + b + c)^2)/(ab + bc + ca) - 2]. (_Princeton._)
1. Solve for p: 2^(p - 3) = 128.
2. Solve for t: t^(3/2) = -27.
3. Find the square root of 8114.4064. What, then, is the square root
of .0081144064? of 811440.64? From any of the above can you
determine the square root of .081144064?
4. The H. C. F. of two expressions is a(a - b), and their L. C. M. is
a^2b(a + b)(a - b). If one expression is ab(a^2 - b^2), what is the
other?
5. Solve (short method):
5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0.
6. Solve
2/m - 3/n + 10/p = -3,
4/m + 5/p + 6/n = 15,
1/m - 1/n + 5/p = -1/2.
7. Simplify
21[2/3]^(1/2) - 5[4/5]^(1/2) + 6[4-1/6]^(1/2)
- 10[3-1/5]^(1/2) + (40/3)[11-1/4]^(1/2).
8. Does [16 x 25]^(1/2) = 4 x 5? Does [16 + 25]^(1/2) = 4 + 5?
9. Write the fraction 5/(4 + 2[3^(1/2)]) with rational denominator,
and find its value correct to two decimal places.
10. Simplify
[{([p + [p^2 - q]^(1/2)]/2)^(1/2)
+ ([p - [p^2 - q]^(1/2)]/2)^(1/2)}^2]/[p + q^(1/2)].
(_Princeton._)
1. Rationalize the denominator of
{6^(1/2) + 3^(1/2) - 3[2^(1/2)]}/{6^(1/2) - 3^(1/2) + 3[2^(1/2)]}.
(_Univ. of Cal._)
2. Simplify [2^(n + 4) - 2(2^n)]/[2(2^(n + 3))]. (_Univ. of Penn._)
3. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)],
when x = 2. (_Cornell._)
4. Find the value of x if
x^(6/5) = y^4,
y^(2/3) = 9. (_M. I. T._)
5. A fisherman told a yarn about a fish he had caught. If the fish
were half as long as he said it was, it would be 10 inches more
than twice as long as it is. If it were 4 inches longer than it
is, and he had further exaggerated its length by adding 4 inches,
it would be 1/5 as long as he now said it was. How long is the
fish, and how long did he first say it was? (_M. I. T._)
6. The force _P_ necessary to lift a weight _W_ by means of a certain
machine is given by the formula
P = a + bW,
where _a_ and _b_ are constants depending on the amount of friction
in the machine. If a force of 7 pounds will raise a weight of 20
pounds, and a force of 13 pounds will raise a weight of 50 pounds,
what force is necessary to raise a weight of 40 pounds? (First
determine the constants _a_ and _b_.) (_Harvard._)
7. Reduce to the simplest form:
[[4/[2^(n + 2)]]^(1/n);
[ax(a^(-1)x - ax^(-1))]/[x^(2/3) - a^(2/3)].
8. Determine the H. C. F. and L. C. M. of (xy - y^2)^3 and y^3 - x^2y.
(_College Entrance Board._)
1. Simplify (a - 8m)/(a^(1/3) - 2m^(1/3)) - 2a^(1/3)m^(1/3).
2. Simplify, writing the result with rational denominator:
([a^(1/2) + (1)/(x^(-1/2))]^2 - [(1)/(a^(-1/2)) - x^(1/2)]^2)
/ (x + [a^2 + x^2]^(1/2)). (_M. I. T._)
3. Find [7 - 48^(1/2)]^(1/2).
4. Expand ([a^3]^(1/2) - [b^5]^(1/2))^5.
5. Expand and simplify (1 - 2[3^(1/2)] + 3[2^(1/2)])^2.
6. Solve the simultaneous equations
x ^(-1/2) + 2y^(-1/2) = 7/6,
2x^(-1/2) - y^(-1/2) = 2/3. (_Yale._)
7. Find to three places of decimals the value of
{[(a + b)^(-1/3)]/[(11a + b^2)^(1/6)]
. [({a^3 - b^3)^(-1/2)]/[(a - b)^(1/2)]}^(1/2),
when a = 5 and b = 3. (_Columbia._)
8. Show that (10 - 4[5^(1/2)])/(5 + 3[5^(1/2)]) is the negative of
the reciprocal of (10 + 4[5^(1/2)])/(5 - 3[5^(1/2)]).
(_Columbia._)
9. Solve and check {5}/{[3x + 2]^(1/2)} = [3x + 2]^(1/2) + [3x - 1]^(1/2).
10. Assuming that when an apple falls from a tree the distance
(S meters) through which it falls in any time (t seconds) is
given by the formula S = (1/2)gt^2 (where g = 9.8), find to two
decimal places the time taken by an apple in falling 15 meters.
(_College Entrance Board._)
Excellent practice may be obtained by solving the ordinary formulas used
in arithmetic, geometry, and physics _orally, for each letter in turn_.
ARITHMETIC
p = br
i = prt
a = p + prt
GEOMETRY
K = (1/2) bh
K = bh
K = (a^2)/4 3^(1/2)
K = (1/2) (b + b') h
K = [pi] R^2
C = 2 [pi] R
K = [pi] R L
S = 4 [pi] R^2
V = [pi] R^2 H
V = (1/3) [pi] R^2 H
V = (4/3) [pi] R^3
S = ([pi] R^2 E)/(180)
C/(C') = R/(R')
K/(K') = (R^2)/(R'^2)
PHYSICS
v = gt
s = (1/2) gt^2
s = (v^2)/(2g)
C = E/R
E = (wv^2)/(2g)
e = (4Pl^3)/(bh^3 m)
E = (mv^2)/(2)
t = [pi] [l/g]^(1/2)
F = (mV^2)/(r)
mh = (mv^2)/(2g)
R = gs/(g + s)
E = (4n^2l^2w)/(g)
C = (5/9)(F - 32)
QUADRATIC EQUATIONS
1. Define a quadratic equation; a pure quadratic; an affected (or
complete) quadratic; an equation in the quadratic form.
2. Solve the pure quadratic (7)/(3S^2) - (11)/(9S^2) = 5/6.
Review the first (or usual) method of completing the square. Solve by it
the following:
3. x^2 + 10x = 24.
4. 2x^2 - 5x = 7.
5. (x - 1)/2 + 2/(x - 1) = 2-1/2.
6. ax^2 + bx + c = 0.
Review the solution by factoring. Solve by it the following:
7. x^2 + 8x + 7 = 0.
8. 24x^2 = 2x + 15.
9. 3 = 10x - 3x^2.
10. -7 = 6x - x^2.
Solve, by factoring, these equations, which are not quadratics:
11. x^4 = 16.
12. x^3 = 8.
13. x^3 = x.
Review the solution by formula. Solve by it the following:
14. 5x^2 - 6x = 8.
15. (1/2)(x + 1) - (x/3)(2x - 1) = -12.
16. x^2 + 4ax = 12a^2.
17. 3x^2 = 2rx + 2r^2.
Solve graphically:
18. x^2 - 2x - 8 = 0.
19. x^2 + x - 2 = 0.
~Reference:~ The chapter on Quadratic Equations in any algebra (first
part of the chapter).
1. Solve by three methods--formula, factoring, and completing the
square: x^2 + 10x = 24.
Review equations in the quadratic form and solve:
2. x^4 - 5x^2 = -4.
3. 2[x^(-2)]^(1/3) - 3[x^(-1)]^(1/3) = 2.
4. (x + 3)/(x - 3) + 6 = 5[(x + 3)/(x - 3)]^(1/2).
(Let y = [(x + 3)/(x - 3)]^(1/2) and substitute.)
5. 3x^2 - 4x + 2[3x^2 - 4x - 6]^(1/2) = 21.
6. x^2 + 5x - 5 = (6)/(x^2 + 5x).
Solve and check:
7. [x + 7]^(1/2) + [3x - 2]^(1/2) = (4x + 9)/([3x - 2]^(1/2)).
8. [x^2 - 5]^(1/2) + 6/[[x^2 - 5]^(1/2)] = 5.
9. (10w)/([10w - 9]^(1/2)) - [10w + 2]^(1/2) = 2/([10w - 9]^(1/2)).
Give results by inspection:
10. (a^(1/2) + b^(1/2))(a^(1/2) - b^(1/2)).
11. ([10 + 19^(1/2)]^(1/2))([10 - 19^(1/2)]^(1/2)).
12. How many gallons each of cream containing 33% butter fat and
milk containing 6% butter fat must be mixed to produce 10
gallons of cream containing 25% butter fat?
13. I have $6 in dimes, quarters, and half-dollars, there being 33
coins in all. The number of dimes and quarters together is ten
times the number of half-dollars. How many coins of each kind
are there? (_College Entrance Board._)
~Reference:~ The last part of the chapter on Quadratic Equations in
any algebra.
THE THEORY OF QUADRATIC EQUATIONS
~I. To find the sum and the product of the roots.~
The general quadratic equation is
ax^2 + bx + c = 0. (1)
Or, x^2 + (b/a)x + c/a = 0. (2)
To derive the formula, we have by transposing
x^2 + (b/a)x = -c/a.
Completing the square,
x^2 + (b/a)x + [b/2a]^2 = (b^2)/(4a^2) - c/a = (b^2 - 4ac)/(4a^2).
Extracting square root, x + b/2a = [+-[b^2 - 4ac]^(1/2)]/(2a).
Transposing, x = -b/2a +- [[b^2 - 4ac]^(1/2)]/(2a).
Hence, x = [-b +- [b^2 - 4ac]^(1/2)]/(2a).
These two values of x we call _roots_.
For convenience represent them by r_1 and r_2.
Hence, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a).
r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).
---------------------------------------------
Adding, r_1 + r_2 = -(2b)/(2a) = -b/a. (3)
Also, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a).
r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).
-------------------------------------------
Multiplying, r_1 r_2 = (b^2)/(4a^2) - (b^2 - 4ac)/(4a^2)
= (b^2 - b^2 + 4ac)/(4a^2)
= (4ac)/(4a^2) = c/a. (4)
Hence we have shown that
r_1 + r_2 = -b/a,
and r_1 r_2 = c/a.
Or, referring to equation (2) above, we have the following rule:
_When the coefficient of x^2 is unity, the sum of the roots is
the coefficient of x with the sign changed; the product of the
roots is the independent term._
EXAMPLES:
1. x^2 - 9x + 21 = 0.
Sum of the roots = 9.
Products of the roots = 21.
2. 3x^2 - 7x - 18 = 0.
Sum of the roots = 7/3.
Product of the roots = -6.
3. -21x = 17 - 4x^2.
Sum of the roots = 21/4.
Product of the roots = -17/4.
~II. To find the nature or character of the roots.~
As before, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a),
r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).
The [b^2 - 4ac]^(1/2) determines the _nature_ or _character_ of the
roots; hence it is called the _discriminant_.
~If b^2 - 4ac is positive, the roots are real, unequal, and either
rational or irrational.~
~If b^2 - 4ac is negative, the roots are imaginary and unequal.~
~If b^2 - 4ac is zero, the roots are real, equal, and rational.~
EXAMPLES:
1. x^2 - 4x + 2 = 0.
[b^2 - 4ac]^(1/2) = [16 - 8]^(1/2) = 8^(1/2).
Therefore: The roots are real, unequal, and irrational.
2. x^2 - 4x + 6 = 0.
[b^2 - 4ac]^(1/2) = [16 - 24]^(1/2) = -8^(1/2).
Therefore: The roots are imaginary and unequal.
3. x^2 - 4x + 4 = 0.
[b^2 - 4ac]^(1/2) = [16 - 16]^(1/2) = 0^(1/2).
Therefore: The roots are real, equal, and rational.
~III. To form the quadratic equation when the roots are given.~
Suppose the roots are 3, -7.
Then, x = 3, Or, x - 3 = 0,
x = -7. x + 7 = 0.
-------------------
Multiplying to get a quadratic, (x - 3)(x + 7) = 0.
Or, x^2 + 4x - 21 = 0.
_Or_, use the sum and product idea developed on the preceding page. The
coefficient of x^2 must be unity.
Add the roots and change the sign to get the coefficient of x.
Multiply the roots to get the independent term.
Therefore: The equation is x^2 + 4x - 21 = 0.
In the same way, if the roots are [2 + 3^(1/2)]/7, [2 - 3^(1/2)]/7, the
equation is
x^2 - (4/7)x + 1/49 = 0.
Find the sum, the product, and the nature or character of the roots of
the following:
1. x^2 - 7x + 12 = 0.
2. 9x^2 - 6x + 1 = 0.
3. x^2 + 2x + 9734 = 0.
4. 16 + 5/x = 17/(x^2).
5. (x - 8)/(x - 3) = x.
6. (x + 7)(x - 6) = 70.
7. x^2 - x(2)^(1/2) = 3.
8. pr^2 + qr + s = 0.
Form the equations whose roots are:
9. 5, -3.
10. 2/3, 5/3.
11. c + d, c - d.
12. -3, -5.
13. [2 +- -3^(1/2)]/5.
14. 8/3 + (2/3)37^(1/2), 8/3 - (2/3)37^(1/2).
15. [-2 +- -2^(1/2)]/2.
16. Solve x^2 - 3x + 4 = 0. Check by substituting the values of x;
then check by finding the sum and the product of the roots.
Compare the amount of labor required in each case.
17. Solve (x - 3)(x + 2)(x^2 + 3x - 4) = 0.
18. Is e^(4z) + 2e^(3z) + e^(2z) + 2e^z + 2 + e^(-2z) a perfect
square?
19. Find the square root (short method):
(x^2 - 1)(x^2 - 3x + 2)(x^2 - x - 2).
20. Solve (1.2x - 1.5)/(1.5) + (.4x + 1)/(.2x - .2) = (.4x + 1)/(.5).
21. The glass of a mirror is 18 inches by 12 inches, and it has a
frame of uniform width whose area is equal to that of the glass.
Find the width of the frame.
OUTLINE OF SIMULTANEOUS QUADRATICS
~Simultaneous Quadratics~
CASE I.
One equation linear.
The other quadratic.
2x + y = 7,
x^2 + 2y^2 = 22.
METHOD: Solve for x as in terms of y, or _vice versa_, in the
linear and substitute in the quadratic.
CASE II.
Both equations homogeneous and of the second degree.
x^2 - xy + y^2 = 39,
2x^2 - 3xy + 2y^2 = 43.
METHOD: Let y = vx, and substitute in both equations.
ALTERNATE METHOD: Solve for x in terms of y in one equation and substitute in the other.
CASE III.
Any two of the quantities
x + y
x^2 + y^2
xy
x - y
x^3 + y^3
x^3 - y^3
x^2 + xy + y^2
x^2 - xy + y^2
given.
x + y = 5,
x^2 - xy + y^2 = 7.
METHOD: Solve for x + y and x - y; then add to get x, subtract
to get y.
CASE IV.
Both equations symmetrical or symmetrical except for sign.
Usually one equation of high degree, the other of the first
degree.
x^5 + y^5 = 242,
x + y = 2.
METHOD: Let x = u + v and y = u - v, and substitute in both
equations.
~Special Devices~
I. Consider some compound quantity like xy, [x - y]^(1/2),
[xy]^(1/2), x/y, etc., as the unknown, at first. Solve
for the compound unknown, and combine the resulting
equation with the simpler original equation.
x^2 y^2 + xy = 6,
x + 2y = -5.
II. Divide the equations member by member. Then solve by
Case I, II, or III.
x^3 - y^3 = 152,
x - y = 2.
III. Eliminate the quadratic terms. Then solve by Case I,
II, or III.
xy + x = 15,
xy + y = 16.
SIMULTANEOUS QUADRATICS
Solve:
1. x + y = 7,
x^2 + 4xy = 57.
2. 2x^2 = 46 + y^2,
xy + y^2 = 14.
3. x^2 + y^2 = 25,
x + y = 1.
4. x^4 + y^4 = 2,
x - y = 2.
5. x^3 + y^3 = 28,
x + y = 4.
6. x^2 y^2 + xy - 12 = 0,
x + y = 4.
7. 2xy - x + 2y = 16,
3xy + 2x - 4y = 10.
8. (3x - 2y)(2x - 3y) = 26,
x + 1 = 2y.
9. 4x^2 + 3xy + 2y^2 = 18,
3x^2 + 2xy - y^2 = 3.
10. x^5 + y^5 = 242,
x + y = 2.
11. x - y + [x - y]^(1/2) = 6,
xy = 5.
12. 4x^2 - x + y = 67,
3x^2 - 3y = 27.
13. x - y - [x - y]^(1/2) = 2,
x^3 - y^3 = 2044. (_Yale._)
14. x^2 + xy + x = 14,
y^2 + xy + y = 28. (_Princeton._)
15. x^2 + y^2 = 13,
y^2 = 4(x - 2). Plot the graph of each equation. (_Cornell._)
16. x^2 + y^2 = xy + 37,
x + y = xy - 17. (_Columbia._)
_In grouping the answers, be sure to associate each value of x with the
corresponding value of y._
17. The course of a yacht is 30 miles in length and is in the shape of
a right triangle one arm of which is 2 miles longer than the
other. What is the distance along each side?
~Reference:~ The chapter on Simultaneous Quadratics in any algebra.
RATIO AND PROPORTION
1. Define ratio, proportion, mean proportional, third proportional,
fourth proportional.
2. Find a mean proportional between 4 and 16; 18 and 50; 12m^2n
and 3mn^2.
3. Find a third proportional to 4 and 7; 5 and 10; a^2 - 9 and a - 3.
4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14.
5. Write out the proofs for the following, stating the theorem in
full in each case:
(_a_) The product of the extremes equals etc.
(_b_) If the product of two numbers equals the product of two other
numbers, either pair etc.
(_c_) Alternation.
(_d_) Inversion.
(_e_) Composition.
(_f_) Division.
(_g_) Composition and division.
(_h_) In a series of equal ratios, the sum of the antecedents is to
the sum of the consequents etc.
(_i_) Like powers or like roots of the terms of a proportion etc.
6. If x : m :: 13 : 7, write all the possible proportions that can be
derived from it. [See (5) above.]
7. Given rs = 161m; write the eight proportions that may be derived
from it, and quote your authority.
8. (_a_) What theorem allows you to change any proportion into an
equation?
(_b_) What theorem allows you to change any equation into a
proportion?
9. If xy = rg, what is the ratio of x to g? of y to r? of y to g?
10. Find two numbers such that their sum, difference, and the sum of
their squares are in the ratio 5 : 3 : 51. (_Yale._)
~Reference:~ The chapter on Ratio and Proportion in any algebra.
An easy and powerful method of proving four expressions in proportion is
illustrated by the following example:
Given a : b = c : d;
prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.
Let a/b = r. Therefore a = br.
Also c/d = r. Therefore c = dr.
Substitute the value of a in the first ratio, and c in the second:
Then
(3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r)
= [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).
Also
(3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r)
= [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).
Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2).
Axiom 1.
Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.
If a : b = c : d, prove:
1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2.
2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2.
3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd.
4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d.
5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c.
6. The second of three numbers is a mean proportional between the
other two. The third number exceeds the sum of the other two by 20;
and the sum of the first and third exceeds three times the second
by 4. Find the numbers.
7. Three numbers are proportional to 5, 7, and 9; and their sum is 14.
Find the numbers. (_College Entrance Board._)
8. A triangular field has the sides 15, 18, and 27 rods, respectively.
Find the dimensions of a similar field having 4 times the area.
~ARITHMETICAL PROGRESSION~
1. Define an arithmetical progression.
Learn to derive the three formulas in arithmetical progression:
l = a + (n - 1)d,
S = (n/2)(a + l),
S = (n/2)[2a + (n - 1)d].
2. Find the sum of the first 50 odd numbers.
3. In the series 2, 5, 8, ..., which term is 92?
4. How many terms must be taken from the series 3, 5, 7, ..., to
make a total of 255?
5. Insert 5 arithmetical means between 11 and 32.
6. Insert 9 arithmetical means between 7-1/2 and 30.
7. Find x, if 3 + 2x, 5 + 6x, 9 + 5x are in A. P.
8. The 7th term of an arithmetical progression is 17, and the 13th
term is 59. Find the 4th term.
9. How can you turn an A. P. into an equation?
10. Given a = -5/3, n = 20, S = -5/3, find d and l.
11. Find the sum of the first n odd numbers.
12. An arithmetical progression consists of 21 terms. The sum of the
three terms in the middle is 129; the sum of the last three terms
is 237. Find the series. (Look up the short method for such
problems.) (_Mass. Inst. of Technology._)
13. B travels 3 miles the first day, 7 miles the second day, 11 miles
the third day, etc. In how many days will B overtake A who started
from the same point 8 days in advance and who travels uniformly 15
miles a day?
~Reference:~ The chapter on Arithmetical Progression in any algebra.
~GEOMETRICAL PROGRESSION~
1. Define a geometrical progression.
Learn to derive the four formulas in geometrical progression:
{ I. l = ar^(n - 1).
{II. S = (ar^n - a)/(r - 1).
{III. S = (rl - a)/(r - 1).
{ IV. S_{[infinity]} = (a)/(1 - r).
2. How many terms must be taken from the series 9, 18, 36, ... to
make a total of 567?
3. In the G. P. 2, 6, 18, ..., which term is 486?
4. Find x, if 2x - 4, 5x - 7, 10x + 4 are in geometrical progression.
5. How can you turn a G. P. into an equation?
6. Insert 4 geometrical means between 4 and 972.
7. Insert 6 geometrical means between 5/16 and 5120.
8. Given a = -2, n = 5, l = -32; find r and S.
9. If the first term of a geometrical progression is 12 and the sum to
infinity is 36, find the 4th term.
10. If the series 3-1/3, 2-1/2, ... be an A. P., find the 97th term.
If a G. P., find the sum to infinity.
11. The third term of a geometrical progression is 36; the 6th term is
972. Find the first and second terms.
12. Insert between 6 and 16 two numbers, such that the first three of
the four shall be in arithmetical progression, and the last three
in geometrical progression.
13. A rubber ball falls from a height of 40 inches and on each rebound
rises 40% of the previous height. Find by formula how far it falls
on its eighth descent. (_Yale._)
~Reference:~ The chapter on Geometrical Progression in any algebra.
~THE BINOMIAL THEOREM~
1. Review the Binomial Theorem laws. (See Involution.)
Expand:
2. (b - n)^7.
3. (x + x^(-1))^5.
4. [a/x - x/a]^6.
5. [x/2y - [xy]^(1/2)]^5.
6. (x^2 - x + 2)^3.
7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4.