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Section 1.6 Solving Quadratic Equations NOTES Precalculus A quadratic equation has a standard form of ax2 + bx + c = 0. a.) For this equation, what letter represents the variable? b.) For this equation, what letters represent the coefficients? c.) For this equation, what letter represents the constant term? d.) What is a root or solution of a quadratic equation? Factoring: Using Example 1 as a guide, solve (2x + 5) (x – 2) = -7 by factoring. Completing the Square: When completing the square, what does one side of the equation become? Using Example 2 as a guide, solve 2x2 – 16x – 40 = 0 by completing the square. The Quadratic Formula What are the roots (solutions for x) for ax2 + bx + c = 0 as an equation: Using Example 3 as a guide, solve 5x2 – 5x + 2 = 0 using the quadratic formula. b b2 4ac : 2a If b2 – 4ac is negative, will the result be real or imaginary? Looking at the quadratic formula, x If b2 – 4ac is positive, will the result be real or imaginary? If b2 – 4ac is not zero, will there be one or two solutions? If b2 – 4ac is zero, will there be one or two solutions? Choosing a Method of Solution What formula can be used to solve any quadratic equation? If a, b, and c are integers and b2 – 4ac is a perfect square, what is the best method to use? If the equation has the form x2 + (even number)x + constant = 0 what is the best method to use? Section 1.6 Solving Quadratic Equations Assignment Precalculus Solve by factoring. 1.) 3x2 – 4x – 7 = 0 2.) (2x – 3) (x + 4) = 6 For questions 3 – 5 solve by completing the square. 3.) x2 – 10x = 1575 4.) x2 + 6x + 10 = 0 5.) y2 + 10y + 35 = 0 For questions 6 – 10 solve using the quadratic formula. 6.) 5x2 + 2x – 1 = 0 7.) 3t2 = 12t – 15 8.) 5u2 + 2 = 5u 9.) 10.) 4 v6 v v4 4 3z z z 3 For questions 11 – 13 solve by whichever method seems easiest. Give both real and imaginary roots. Be sure not to lose or gain roots. 11.) 8x2 = 7 – 10x 12.) (4x + 7) (x – 1) = 2(x – 1) t2 1 t 5 13.) t2 3 t2 For problems 14 and 15, DE is parallel to BC . Find the value of x. A 14.) x D x+3 B 8 E 2x C A 15.) x+6 D 6x E x 7x + 1 B 16.) For 4x2 + 8x + k = 0: a.) What is the discriminant? b.) For what values of k will the equation have a double root? c.) For what values of k will the equation have two real roots? d.) For what values of k will the equation have imaginary roots? 17.) Use the quadratic formula to solve ix2 – 3x – 2i = 0. 18.) Derive the quadratic formula (Hint: solve the general quadratic equation ax2 + bx + c = 0, (a not equal to zero) by completing the square). C