Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square Perfect Square Trinomials Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36 Creating a Perfect Square Trinomial In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____ Find the constant term by

squaring half the coefficient of the linear term. (14/2)2 X2 + 14x + 49 Perfect Square Trinomials Create perfect square trinomials. x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___ 100 4 25/4 Solving Quadratic Equations by Completing the Square

Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation 2 x 8 x 20 0 2 x 8 x 20 Solving Quadratic Equations by Completing the Square Step 2:

Find the term that completes the square on the left side of the equation. Add that term to both sides. 2 x 8x =20 + 1 (8) 4 then square it, 4 2 16 2 x 8 x 16 20 16 2 Solving Quadratic Equations by Completing the Square Step 3:

Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. x 8 x 16 20 16 2 Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

2 ( x 4) 36 ( x 4) 6 Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilitie s and solve x 4 6 x 4 6 and x 4 6 x 10 and x=2 Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move quadratic

term, and linear term to left side of the equation, the constant to the right side of the equation. 2 2 x 7 x 12 0 2 2 x 7 x 12 Solving Quadratic Equations by Completing the Square 2 x 7 x =-12 + Step 2: Find the term 2x 7x 12 that completes the

square on the left side of 2 2 2 2 2 the equation. Add that term to both sides. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first. 7 x x 6 2

2 1 7 7 ( ) then square it, 2 2 4 7 49 49 x x 6 2 16 16 2 2 49 7

16 4 Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. 7 49 49 x x 6

2 16 16 2 2 7 96 49 x 4 16 16 2 7 47 x 4 16

Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side 7 2 47 (x ) 4 16 7 47 ( x ) 4 4 7 i 47 x 4

4 7 i 47 x 4 Solving Quadratic Equations by Completing the Square Try the following examples. Do your work on your paper and then check your answers. 2 1. 9, 7 2 2.(6, 14) 2 3. 3,8 1. x 2 x 63 0 2. x 8 x 84 0

3. x 5 x 24 0 2 4. x 7 x 13 0 5. 3 x 2 5 x 6 0 7 i 3 4. 2 5 i 47 5. 6