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)).