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Math 1014: Precalculus with Transcendentals Ch. 1: Equations and Inequalities Sec. 1.5: Quadratic Equations I. Quadratic Equation A. Definition x is an equation that can be written in the general form ax + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0 . A quadratic equation is A quadratic equation in 2 a second-degree polynomial equation. B. Methods for Solving Quadratic Equations 1. Solving Quadratic Equations by Factoring a. Zero-Product Principle If the product of 2 algebraic expressions is zero, then at least one of the factors is equal to zero. If AB = 0, then A = 0 or B = 0. b. Steps 1) Put equation in the form, ax 2 + bx + c = 0 . 2) Factor completely. 3) Set each factor equal to 0 and solve for x. 4) Check solutions in original equation. c. Examples: Solve each equation by factoring. 1) x 2 = −11x − 10 2) 4x 2 − 13x = −3 2. Solving Quadratic Equations by the Square Root Property a. The Square Root Property If u is an algebraic expression and d is a nonzero number, then u 2 = d has exactly two solutions: If u 2 = d, then u = ± d . b. Steps 1) Put equation in the form, u2 = d . 2) Solve 3) Check solutions in original equation. c. Examples: Solve each equation by the square root property. 1) 3x 2 − 1 = 47 2) (x − 3)2 − 36 = 0 3. Solving Quadratic Equations by Completing the Square a. Completing the Square 2 ⎛ b⎞ If x + bx is a binomial then adding ⎜ ⎟ will result in a perfect square trinomial. ⎝ 2⎠ 2 2 2 b⎞ ⎛ b⎞ ⎛ x + bx + ⎜ ⎟ = ⎜ x + ⎟ . ⎝ 2⎠ ⎝ 2⎠ 2 b. Steps 1) Put equation in the form, x 2 + bx = c . 2 ⎛ b⎞ 2) Add ⎜ ⎟ to both sides of the equation. ⎝ 2⎠ 2 3) Factor left hand side of equation, b⎞ ⎛ ⎜⎝ x + ⎟⎠ and simplify right hand side. 2 4) Take square root of both sides. Make sure to take 5) Solve each resulting equation for x. 6) Check solutions in original equation. c. Examples: Solve each equation by completing the square. 1) x 2 + 4x = 12 2) x 2 − 3x − 5 = 0 ± roots of RHS. 4. Solving Quadratic Equations by Quadratic Formula a. The Quadratic Formula The solution of a quadratic equation in general form a ≠ 0 , are given by the quadratic formula: ax 2 + bx + +c = 0 , with −b ± b 2 − 4ac x= 2a b. Number of Solutions 1) The value of the discriminant, b 2 − 4ac , determines the number and kind of solutions. b 2 − 4ac > 0 : Two unequal real solutions 2 b) b − 4ac = 0 : One real solution 2 c) b − 4ac < 0 : No real solutions; Two complex solutions a) c. Examples: Solve each equation using the quadratic formula. 1) x 2 − 6x + 9 = 0 2) 3x 2 − 4x = 4 3) x 2 − 2x = −17 4) 5x 2 + x − 2 = 0