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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 2. 𝑖√11 3. 𝒊√𝟕 𝟑 Solve each equation by taking square roots. Give exact answers in reduced radical form. Tell whether the solutions are real or imaginary. 4. 7𝑥 2 − 12 = 0 5. 𝑥 2 + 9 = 3 6. 2(𝑥 2 − 1) = 𝑥 2 − 3 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 object’s 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? 8. An acorn fell from a branch 20 feet high and landed on a branch 7 feet high. For how many seconds was the acorn falling? 3.2 Complex Numbers Simplify each expression. 1. √−25 2. 3√−49 3. −√−81 1 Add, subtract, or multiply the complex numbers. 4. 4𝑖 + (2 + 8𝑖) 5. (2 − 7𝑖) − (5 − 3𝑖) 6. (3 + 𝑖)(1 − 4𝑖) 3.3 Finding Complex Solutions of Quadratic Equations Solve each by completing the square. Express answers in reduced radical form. 1. 𝑥 2 + 2𝑥 = 3 2. 3𝑥 2 = 8 + 10𝑥 3. −3𝑥 2 + 7𝑥 = −30 4. 4𝑥 2 = −12𝑥 + 4 Use the discriminant to determine the number and type of solutions. (real, non-real) 5. 𝑥 2 − 3𝑥 = −8 6. 𝑥 2 + 4𝑥 = −3 7. 2𝑥(𝑥 − 6) = −18 8. The function ℎ(𝑡) = −9.8𝑡 2 + 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 root of a batting cage with a maximum height of 64.5 m? Use the discriminant to explain your answer. Solve using the Quadratic Formula. Express answers in reduced radical form. 9. 𝑥 2 + 10𝑥 = −9 10. 𝑥 2 + 2𝑥 = −4 11. 𝑥 2 + 5𝑥 = 3 12. 2𝑥 2 + 7𝑥 + 10 = 0 4.1 Circles Write the standard form equation of each circle. 1. The circle with center C(-3, 2) and radius r = 4. 2. The circle with center C(-4, -3) and containing the point P(2, 5). Graph each circle after writing the equation in Standard Form. Identify its center and radius. 3. 4𝑥 2 + 4𝑦 2 + 8𝑥 − 16𝑦 + 11 = 0 Center: ____________Radius: _________ 4. 𝑥 2 + 𝑦 2 + 4𝑥 + 6𝑦 + 4 = 0 Center: ____________Radius: _________