A Review of Algebra

Part 2

Chapter 24,602 wordsPublic domain

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 coördinate 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 × 3 - (2 · 7^2)/(14) ÷ 7 × 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 × 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 × a^(-m) = a^(m - m) = a^0 = 1.

Hence, a^m × 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 × 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 × 5, 3 × 5^0, 3^0 × 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) × [(c^3d)/(ab^3)]^(1/3) × [(a^(1/3)c)/(b^(1/4)d^(5/12))]^2.

10. [(a^(-4))/(b^(-2)c)]^(-3/4) × [(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 × (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) × [((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 × 25]^(1/2) = 4 × 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.