Algebra 2 Module 3 and 4.1 Review

**H. Algebra 2** **Module 3 and 4.1 Review** **Name:** **Date:** **Hour:** **3.1 Solving Quadratic Equations by Taking Square Roots** Find the square of each imaginary number. 1. (4i)(4i) = 16i² = -16 2. (√11i)² = 11i² = -11 3. (√7/3)² = 7i²/9 = -7/9 Solve each equation by taking square roots. Give exact answers in reduced radical form. Tell whether the solutions are real or imaginary. 4. 7x² - 12 = 0 7x² = 12 x² = 12/7 x = ±√(12/7) = ±2√3/√7 5. x² + 9 = 3 x² = -6 x = ±i√6 6. 2(x² - 1) = x² - 3 2x² - 2 = x² - 3 x² = -1 x = ±i For 7-8, use the equation h(t) = h₀ - 16t², that models the height of an object in feet after time t in seconds, and h₀ is the initial height in feet. 7. A carpenter dropped a hammer from a rooftop 48 feet above ground. How long did it take for the hammer to hit the ground? h(t) = 0 0 = 48 - 16t² 48 = 16t² t² = 3 t = ±√3 ≈ ±1.73 seconds 8. An acorn fell from a branch 20 feet high. For how many seconds was the acorn falling? h(t) = 20 - 16t² 0 = 20 - 16t² 20 = 16t² t² = 5/4 t = ±√(5/4) ≈ ±1.12 seconds **3.2 Complex Numbers** Simplify each expression. 1. √-25 = 5i 2. 3√-49 = 3 * 7i = 21i 3. -√-81 = -9i **Add, subtract, or multiply the complex numbers.** 1. 4i + (2 + 8i) = 2 + 12i 2. (2 - 7i) - (5 - 3i) = -3 - 4i 3. (3 + i)(1 - 4i) = 3 - 12i + i - 4i² = 7 - 11i **3.3 Finding Complex Solutions of Quadratic Equations** Solve each by completing the square. Express answers in reduced radical form. 1. x² + 2x = 3 x² + 2x + 1 = 4 (x + 1)² = 4 x + 1 = ±2 x = -1 ± 2 x = -3, 1 2. 3x² - 10x = 8 3x² - 10x + 25/9 = 8 + 25/9 (x - 5/3)² = 97/9 x - 5/3 = ±√(97/9) x = (5 ± √97)/3 3. 4x² + 12x = 4 x² + 3x = 1 x² + 3x + 9/4 = 1 + 9/4 (x + 3/2)² = 13/4 x + 3/2 = ±√(13/4) x = (-3 ± √13)/2 **Use the discriminant to determine the number and type of solutions. (real, non-real)** 5. x² - 2x = 8 b² - 4ac = (-2)² - 4(1)(8) = -28 No solution 6. x² + 4x = -3 b² - 4ac = 4² - 4(1)(3) = 4 2 real solutions 7. 2x(x - 6) = -18 2x² - 12x + 18 = 0 b² - 4ac = (-12)² - 4(2)(18) = 0 1 real solution 8. The function h(t) = -9.8t² + 45t + 1.2 models 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. 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. 64.5 = -9.8t² + 45t + 1.2 0 = -9.8t² + 45t - 63.3 **The ball will not reach the roof.** 45² - 4(-9.8)(-63.3) = -456.36 no solution Solve using the Quadratic Formula. Express answers in reduced radical form. 9. x² + 10x = -9 x² + 10x + 9 = 0 (x + 9)(x + 1) = 0 x = -9 or x = -1 10. x² + 2x = -4 x² + 2x + 4 = 0 x = (-2 ± √(2² - 4(1)(4))) / (2(1)) x = (-2 ± 2i√3) / 2 x = -1 ± i√3 11. x² + 5x = 3 x² + 5x - 3 = 0 x = (-5 ± √(5² - 4(1)(-3))) / (2(1)) x = (-5 ± √37) / 2 12. 2x² + 7x + 10 = 0 x = (-7 ± √(7² - 4(2)(10))) / (2(2)) x = (-7 ± √-31) / 4 x = (-7 ± i√31) / 4 **4.1 Circles** Write the standard form equation of each circle. 1. The circle with center C(3, -2) and radius r = 4. (x - 3)² + (y + 2)² = 16 2. The circle with center C(-4, -3) and containing the point P(2, 5). (x + 4)² + (y + 3)² = 100 (2+4)² + (5+3)² = 6² + 8² = 36 + 64 = 100