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Solving Quadratic Equations by Completing the Square The equation must be in standard form: ax 2 + bx + c = 0 For example: Solve the equation by completing the square: 2 x 2 + 12 x + 3 = 0 Therefore a = 2, b = 12, and c = 3 Step 1 If a = 1, then go on to Step 2. Otherwise, divide both sides of the equal sign by a. Since a is not equal to 1. . . 3 2 x 2 12 x 3 0 2 Divide every term by 2. . . ö + + = . . . to get ö x + 6 x + = 0 2 2 2 2 2 Step 2 Get all variable terms on one side of the equation and all constants on the other side. x 2 + 6x + 3 −3 = 0 ö x 2 + 6x = 2 2 Step 3 2 ⎛b⎞ Complete the square for the resulting binomial by adding ⎜ ⎟ to both sides of the equation. ⎝2⎠ 2 2 −3 ⎛6⎞ −3 ⎛6⎞ ö x 2 + 6x + ⎜ ⎟ = +⎜ ⎟ 2 2 ⎝2⎠ ⎝2⎠ −3 2 2 2 b. Divide the 6 by 2 in the parentheses ö x + 6 x + (3) = + (3) 2 −3 2 c. Square the resulting 3 and add it to both sides ö x + 6 x + 9 = +9 2 a. b = 6 ö x 2 + 6x = d. Convert 9 to a fraction to add constants on right side of equation ö x e. Add the fractions on the right side of the equation ö The Math Center ■ Valle Verde ■ x 2 + 6x + 9 = Tutorial Support Services 2 + 6x + 9 = − 3 18 + 2 2 15 2 ■ EPCC 1 Step 4 Factor the resulting perfect square trinomial and write it as the square of a binomial. x 2 + 6x + 9 = 15 15 2 ö ( x + 3) = 2 2 Step 5 Use the square root property to solve for x. a. Take the square root of both sides of the equation . . . ö ( x + 3)2 b. Subtract 3 from both sides to isolate the variable ö x = −3 ± c. This leaves the final answers of ö x = −3 + 15 15 . . . to get ö x + 3 = ± 2 2 = 15 2 15 15 and x = −3 − 2 2 Practice Exercises: 1. 3x 2 − 4 x = 4 2. y 2 + 8 y + 18 = 0 3. p2 + p − 2 = 0 4. t 2 − 6t + 3 = 0 Answers: 1. ⎧ 2 ⎫ ⎨− , 2 ⎬ ⎩ 3 ⎭ 2. The Math Center {− 4 + i ■ 2 ,−4 − i 2 Valle Verde ■ } 3. {− 2,1} Tutorial Support Services 4. ■ EPCC {3 + 6 ,3 − 6 } 2