University:
College Entrance Preparation
Course:
ACT Preparation (Mathematics)
Academic year:

2024
Views:

87

Pages:

3
Author:

Campbell Wagner

H. Algebra 2 Module 3 Review Name: Date: Hour: LESSON 3-1 Solving Quadratic Equations by Taking Square Roots For problems 1-3, solve the equation -2x² + 7 = -1 using the indicated method. Show your work. 1. Solve by graphing. y = -2x² + 7 y = -1 [Image of a graph with the parabola y = -2x² + 7 and the line y = -1 intersecting at x = -2 and x = 2] Solution: x = -2, x = 2 2. Solve by factoring. -2x² + 7 = -1 -2x² + 8 = 0 -2(x² - 4) = 0 -2(x + 2)(x - 2) = 0 x + 2 = 0 or x - 2 = 0 x = -2 or x = 2 3. Solve by taking square roots. -2x² + 7 = -1 -2x² = -8 x² = 4 x = ±2 Find the square of each imaginary number: 4. 4i (4i)² = 16(-1) = -16 5. √7i (√7i)² = 7(-1) = -7 6. (4√7i)/3 (4√7i/3)² = (16 * 7)/9 = -112/9 Solve each equation by taking square roots. Tell whether the solutions are real or imaginary. 7. 7x² - 12 = 0 7x² = 12 x² = 12/7 x = ±√(12/7) = ±(2√3)/√7 = ±(2√21)/7 8. x² + 9 = 3 x² = -6 x = ±i√6 9. 2(x² - 1) = (x² - 3) 2x² - 2 = x² - 3 x² = -1 x = ±i Recall the equation for falling objects: h(t) = h₀ - 16t², where h is the height of the object, in feet, at any time t, in seconds, and h₀ is the object's initial height in feet. Use this equation for Problems 10-11. 10. A carpenter dropped a hammer from a rooftop 48 feet high. How long did it take the hammer to hit the ground? -16t² + 48 = 0 -16t² = -48 t² = 3 t = ±√3 ≈ ±1.73 sec 11. An acorn fell from a branch 20 feet high and landed on a branch 7 feet high. How long did it take the acorn to fall? -16t² + 20 = 7 -16t² = -13 t² = 13/16 t ≈ ±0.9 sec LESSON 3-2 Complex Numbers 1. √-25 = 5i 2. 3√-49 = 21i 3. √-81 = -9i Simplify each expression. Write your answer as a complex number. 4. (4i) + (2 + 8i) 2 + 12i 5. (2 + 7i) - (5 - 3i) -3 + 10i 6. (3 + i)(1 - 4i) 7 - 11i LESSON 3-3 Finding Complex Solutions of Quadratic Equations 1. x² + 10x + 9 = 0 x = -9, -1 2. x² + 2x + 4 = 0 x = -1 ± i√3 3. x² + 5x = 3 x = (-5 ± √37)/2 4. 2x² + 7x + 10 = 0 x = (-7 ± i√31)/4 Find the discriminant of each equation. Then determine the number of real or nonreal solutions. 5. x² - 3x + 8 = 0 -23 - 2 nonreal solutions 6. x² + 4x + 3 = 0 4 - 2 real solutions 7. 2x² + 12x + 18 = 0 0 - 1 real solution Solve each equation by completing the square. 11. x² + 2x = 3 (x + 1)² = 4 x = 1, -3 12. 3x² = 8 + 10x 3x² - 10x - 8 = 0 (3x - 4)(x + 2) = 0 x = 4/3, -2 13. x² - 3x = -30 (x - 3/2)² = -81/4 x = (3 ± 9i)/2 14. x² + 7x = 4 (x + 7/2)² = 69/4 x = (-7 ± √69)/2 15. The height in meters of a baseball t seconds after it is hit straight up in the air with a velocity of 45 m/s is given by h = -9.8t² + 45t + 1.2. Will a baseball hit straight up with this velocity hit the roof of a batting cage with a maximum height of 64.5 m? Use the discriminant to explain your answer. [See attached]