Math Algebra Quadratics Review Solutions

Algebra 2 Honors Quadratics Test Review Name: Multiply. Assume that variable exponents represent positive integers. 1. (3y^(3x) + 7)(y^(2x) - 6) = 3y^(5x) - 18y^(3x) + 7y^(2x) - 42 2. (2x + 3)(5x^2 - x + 4) = 10x^3 + 13x^2 + 5x + 12 Solve the following by Factoring. You MUST use Factoring Techniques and NOT FOILING. 3. -3x^4 + 81x = 0 -3x(x^3 - 27) = 0 -3x(x - 3)(x^2 + 3x + 9) = 0 x = 0, 3, (-3 ± 3i√3)/2 4. 9x^2 - 36x = 45 9(x^2 - 4x - 5) = 0 9(x - 5)(x + 1) = 0 x = 5, -1 5. 2x^2 + 3 = -5x 2x^2 + 5x + 3 = 0 (2x + 3)(x + 1) = 0 x = -3/2, -1 6. 3(x^2 - 16) = 4(x^2 - 16) 3x^2 - 4x - 48 = 0 (3x - 4)(x - 12) = 0 x = 4/3, 12 7. x^3 - x^2 - 4x + 4 = 0 x^2(x - 1) - 4(x - 1) = 0 (x - 1)(x^2 - 4) = 0 (x - 1)(x + 2)(x - 2) = 0 x = 1, ±2 8. x^4 - 12x^2 + 27 = 0 (x^2 - 9)(x^2 - 3) = 0 (x + 3)(x - 3)(x^2 - 3) = 0 x = ±3, ±√3 9. 2(x - 3)^2 + (x - 3) = 1 2w^2 + w - 1 = 0 (2w - 1)(w + 1) = 0 w = 1/2, -1 x - 3 = 1/2, -1 x = 7/2, 2 10. 2(6x)(x - 5)^4 = 6(6x)^2(x - 5)^3 2ab^4 - 6a^2b^3 = 0 2ab^3(b - 3a) = 0 12x(x - 5)^3(-17x - 5) = 0 x = 0, 5, -5/17 11. -2x^3 = -16 -2(x^3 - 8) = 0 -2(x - 2)(x^2 + 2x + 4) = 0 x = 2, (-2 ± 2i√3)/2 12. 2(x + 2/3)^2 - 3(x + 2/3) = -1 2w^2 - 3w + 1 = 0 (2w - 1)(w - 1) = 0 w = 1/2, 1 x + 2/3 = 1/2, 1 x = -1/2, 1 Solve the following Inequalities. Use Sign Analysis to verify your solutions and write your answers in Interval Notation. 14. x^2 - 5x + 6 > 0 (x - 2)(x - 3) > 0 x < 2 or x > 3 (-∞, 2) U (3, ∞) 15. 2x^2 + x - 6 ≤ 0 (2x - 3)(x + 2) ≤ 0 -2 ≤ x ≤ 3/2 [-2, 3/2] 16. x + 6 ≥ (x - 2)(x - 3) x + 6 ≥ x^2 - 5x + 6 0 ≥ x^2 - 6x 0 ≥ x(x - 6) x ≤ 0 or x ≥ 6 (-∞, 0] U [6, ∞) 17. √(-9) - √(-10) - √(-12) 3i - i√10 - 2i√3 (-2 + i√10)i 18. √(-9) + √(-10) - √(-25) 3i + i√10 - 5i (-2 + √10)i 19. √(-72x^5) * √(-6y^4) 6x^2y^2i√12x -12x^2y^2√3x 20. √(16x^3)/√(9x) 4x√x / 3√x 4x/3 21. √(-16x)/√(9x^3) 4i√x / 3x√x 4i/3x 22. √(12x^2y)/√(18y) 2i√3x / 3√2 i√2x/3 23. √(2x^2)/√(-4y^2) i√2x / -2iy -√2x/2y 24. √(2x^3)/√(-5y) i√2x² / -i√5y -√(2x²)/√5y 25. √(-8x^2z)/√(-27x^3z) 2i√2x / -3i√3x -2√2/3√3 26-27. Solve the following by Completing the Square. 26. 2x^2 - 10x + 9 = 0 2(x^2 - 5x + 25/4) = 9 - 25/2 2(x - 5/2)^2 = 7/2 (x - 5/2)^2 = 7/4 x - 5/2 = ±√7/2 x = (5 ± √7)/2 27. -4x^2 + 6x = 18 -4(x^2 - 3/2x + 9/16) = 18 - 9/4 -4(x - 3/4)^2 = 63/4 (x - 3/4)^2 = -63/16 x - 3/4 = ±3i√7/4 x = (3 ± 3i√7)/4 28. Rewrite the following equation in Vertex Form by Completing the Square: f(x) = -8x² + 16x - 1 f(x) = -8(x² - 2x) - 1 f(x) = -8(x² - 2x + 1) - 1 + 8 f(x) = -8(x - 1)² + 7 AOS: x = 1 Vertex: (1, 7) Max/Min and Value: 7 x-int: (1 ± √7/2, 0) y-int: (0, -1) Domain: (-∞, ∞) Range: (-∞, 7] 29. Graph the equation y = x². Identify the transformations of the additional equations and graph them as well. This is to be done without a calculator. List the Domain and Range of each graph. a. y = -x² + 5 Trans: Reflect over x-axis, up 5 Domain: (-∞, ∞) Range: (-∞, 5] b. y = 2(x - 2)² - 4 Trans: V. Stretch by 2, Right 2, Down 4 Domain: (-∞, ∞) Range: [-4, ∞) 30-31. Identify all key pieces of information and graph the following quadratics. 30. y = 2x² + 8x - 5 x = -8/2(2) = -2 y = 2(-2)² + 8(-2) - 5 AOS: x = -2 Vertex: (-2, -13) Max/Min Value: -13 x-int: (-4 ± √21)/2 y-int: (0, -5) Domain: (-∞, ∞) Range: [-13, ∞) 31. y = -(1/2)(x + 3)² + 5 x = -3 y = -(1/2)(0 + 3)² + 5 AOS: x = -3 Vertex: (-3, 5) Max/Min Value: 5 x-int: (-3 ± √10, 0) y-int: (0, 1/2) Domain: (-∞, ∞) Range: (-∞, 5] 32-33. Find an equation y = a(x-h)² + k satisfying the given conditions. 32. Vertex (4, 5), contains (1, -4) -4 = a(1-4)² + 5 -9 = 9a a = -1 y = -1(x-4)² + 5 33. x-intercepts are -2 and 8; minimum of y is -25 y = a(x+2)(x-8) -25 = a(3)² a = -25/9 y = -25/9(x+2)(x-8) 34-35. Solve using the Quadratic Formula. 34. 2t² + 4t - 5 = 0 t = (-4 ± √(16+40))/4 t = (-4 ± √56)/4 t = (-2 ± √14)/2 35. x² - √3x - 2 = 0 x = (√3 ± √((-√3)²-4*1*(-2)))/2 x = (√3 ± √11)/2 36-37. Give the value of the discriminant and determine the nature of the roots of each equation. If the equation is factorable, say so.   36. 4x² + 3x + 12 = 3 - 3x 4x² + 6x + 9 = 0 6² - 4(4)(9) = -108; 2 imaginary roots 37. 7x² - 2x - 5 = 0 (-2)² - 4(7)(-5) = 144; 2 rational roots; factorable 38. Find the value(s) of k for which the equation -2x² + 8x + k = 0 has the following roots: A) A real double root, B) Two different real roots, and C) Imaginary Roots: A: Disc = 0 8² - 4(-2)k = 0 64 + 8k = 0 k = -8 B: Disc > 0 64 + 8k > 0 k > -8 C: Disc < 0 8k < -64 k < -8 39-41. Find a quadratic equation with integral coefficients having the given roots. Be sure to use the Sum and Products of Roots Theorem.   39. 2/3, -3/5 Sum: 1/15 Product: -2/5 15x² - x - 6 = 0 40. (1+√2)/3, (1-√2)/3 Sum: 2/3 Product: -1/9 9x² - 6x - 1 = 0 41. 3-i, 3+i Sum: 6 Product: 10 x² - 6x + 10 = 0