Math Quadratics Test Review Key

Quadratics Test Name: Assignment: Algebra 2 Honors Review 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 - 2x^2 + 8x + 15x^2 - 3x + 12 10x^3 + 13x^2 + 5x + 12 Solve 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, -3/2 ± 3√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. 8x^3 - x^4 - 4x = 16 x^4 - 8x^3 + 4x + 16 = 0 (x^2 - 4)(x^2 - 4x + 4) = 0 (x + 2)(x - 2)(x - 2)^2 = 0 x = ±2, 2 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^3 - 9x^2 + 27x = 27 x^3 - 9x^2 + 27x - 27 = 0 (x - 3)^3 = 0 x = 3 9. 2(x - 3)^2 + (x - 3) = 1 2u^2 + u - 1 = 0 (2u - 1)(u + 1) = 0 u = 1/2, -1 x - 3 = 1/2 x - 3 = -1 x = 7/2, 2 10. 2ab(b - 3a) = 0 2ab(b - 3a) = 0 a = 0, b = 0, b = 3a 11. -2x^2 - 16 = 0 -2(x^2 + 8) = 0 x^2 + 8 = 0 x = ±2√2i 12. 3x - 6x^(1/2) - 10 = 0 (3√x + 2)(√x - 5) = 0 √x = -2/3, 5 x = 25/4 Solve the following Inequalities. Use Sign Analysis to verify your solutions and write your answers in Interval Notation. 14. x² + 3x + 6 > 0 (x + 2)(x + 3) > 0 x < -3 or x > -2 (-∞, -3) U (-2, ∞) 15. 2x² + 8x ≤ 0 2x(x + 4) ≤ 0 x = 0, -4 [-4, 0] 16. (x + 1)/(x - 3) ≤ x/(x + 5) (x + 1)(x + 5) ≤ x(x - 3) x² + 6x + 5 ≤ x² - 3x 9x ≤ -5 x ≤ -5/9 (-∞, -5/9] 17. √(-9) * √(-10) * √(-12) 3i * √10 * 2i√3 -6i√30 18. √(-9) + √(-10) - √(-25) 3i + i√10 - 5i (-2 + √10)i 19. √(-72x⁵) * √(-6y⁴) 6x²y²i√12x -12x²y²√3x 20. √(-96x⁴) / √(-6x²) 4x²i√6 / -ix√6 -4x²i 21. √(-108y³) / √(-3y) 6y²i√3 / -iy√3 -6y² 22. √(-75) / √(-10) 5i√3 / i√10 5√3/√10 √15 23. 2√(-2) * √(-5) 2i√2 * i√5 -2i√10 24. 2√(-8) / √(-2) 2i√8 / i√2 4√2 25. √(-27) / √(-12) 3i√3 / 2i√3 3/2 26-27. Solve the following by Completing the Square. 26. 2x² - 10x + 9 = 0 2(x² - 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 = 5/2 ± √7/2 27. -4x² + 6x = 18 -4(x² - 3/2x) = 18 -4(x² - 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 = 3/4 ± 3i√7/4 28. Rewrite the following equation in Vertex Form by Completing the Square: f(x) = -8x² + 16x - 1 and identify all key pieces of information. 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 ± √7/4, 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 4 = 2(-2)² + 8(-2) - 5 -13 = 2(-2)² + 8(-2) - 5 AOS: x = -2 Vertex: (-2, -13) Max/Min Value: -13 x-int: (-4 ± √26)/2 y-int: (0, -5) Domain: (-∞, ∞) Range: [-13, ∞) 31. y = -(1/2)(x + 3)² + 5 -10 = -(1/2)(x + 3)² 10 = (x + 3)² x = -3 ± √10 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, 45) 5 = a(1 - 4)² + 5 0 = 9a a = 0 y = 0(x - 4)² + 5 y = 5 33. x-intercepts are -2 and 8; minimum of y is -25 y = a(x + 2)(x - 8) -25 = a(0 + 2)(0 - 8) -25 = -16a a = 25/16 y = 25/16(x + 2)(x - 8) y = 1/16(x + 2)(x - 8) 34-35. Solve using the Quadratic Formula. 34. 2t(t + 2) = 5 2t² + 4t - 5 = 0 t = (-4 ± √(4² - 4(2)(-5))) / 4 t = (-4 ± √44) / 4 t = (-2 ± √11) / 2 35. x² - x√3 - 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 (-2)² - 4(4)(9) = -144 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 has the following roots: A) A real double root, B) Two different real roots, and C) Imaginary Roots. 8x² + kx + 8 = 0 A: Disc = 0 k² - 4(8)(8) = 0 k² = 256 k = ±16 B: Disc > 0 k² - 4(8)(8) > 0 k² > 256 k < -16 or k > 16 C: Disc < 0 k² - 4(8)(8) < 0 k < -16 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: 2/3 - 3/5 = 1/15 Product: 2/3 * -3/5 = -2/5 15x² - x - 6 = 0 40. (1 + √2)/3, (1 - √2)/3 Sum: 2/3 Product: -1/9 x² - 2x/3 - 1/9 = 0 41. 3 - i, 3 + i Sum: 3 - i + 3 + i = 6 Product: (3 - i)(3 + i) = 9 + 1 = 10 x² - 6x + 10 = 0