Part 29

12. The 8th term of (1 + x[y^(1/2)])^(13).

13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10).

14. The term not containing x in [x^3 - 2/x]^(12).

15. The term containing x^(18) in [x^2 - a/x]^(15).

~Reference:~ The chapter on The Binomial Theorem in any algebra.

~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~

1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of the roots correct to three significant figures. (Harvard.)

2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0. (Sheffield Scientific School.)

3. Solve 2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}). (Yale.)

4. Solve the equation V = (H/3)(B + x + ^(1/2)) for x, taking H = 6, B = 8, and V = 28; and verify your result. (Harvard.)

5. Solve { x : y = 2 : 3, { x^2 + y^2 = 5(x + y) + 2.

6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (Coll. Ent. Board.)

7. Find all values of x and y which satisfy the equations: { x^(1/2) + y^(1/2) = 4, { 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y. (Mass. Inst. of Technology.)

8. If [alpha] and represent the roots of px^2 + qx + r = 0, find [alpha] + , [alpha] - , and [alpha] in terms of p, q, and r. (Princeton.)

9. Form the equation whose roots are 2 + ^(1/2) and 2 - [-3]^(1/2).

10. Determine, without solving, the character of the roots of 9x^2 - 24x + 16 = 0. (College Entrance Board.)

11. If a : b = c : d, prove that a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2). (College Entrance Board.)

12. Given a : b = c : d. Prove that a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d). (Sheffield.)

13. The 9th term of an arithmetical progression is 1/6; the 16th term is 5/2. Find the first term. (Regents.)

Solve graphically:

1. x^2 - x - 6 = 0.

2. x^2 + 3x - 10 = 0.

3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19.

4. What number added to 2, 20, 9, 34, will make the results proportional?

5. Find the middle term of [3a^5 + (b^(3/4))/(2)]^8.

6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2). (Princeton.)

7. A strip of carpet one half inch thick and 29-6/7 feet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately) 22/7 times the diameter. (Harvard.)

8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term? (Yale.)