Factoring Special Cases Factor completely 2 9 x

Factoring Special Cases Factor completely: 2 9 x + 66 x + 21

Difference of Perfect Squares For all numbers a and b, a 2 – b 2 = (a-b) (a+b) Example 1: (x 2 - 25) Since 25 is a perfect square: Therefore: a 2 = x 2 and x a = ___ and b 2 = 25 5 b = ___ Solution: (x 2 - 25) = ( x - 5 ) ( x + 5 )

Example 2: 10 x 2 - 40 2 – 4) 10 (x • First factor out the 10: _____ • Now we have a difference of perfect squares ( x 2 and 4 ) • Note that it has to be a difference so the inner and outer products cancel each other out. Solution: 10 ( x-2 ) ( x+2 )

Perfect Square Trinomials Example 3: x 2 + 12 x + 36 (ax 2 + bx + c) To identify a perfect square trinomial, 1) See if a (1) and c (36) are perfect squares. 2) Multiply square roots of a & c & 2. ________ x 6 x 2 =b 12 3) If the answer 1 equals (12), you have a perfect square trinomial. _________________ x+6 4) Factor by adding/subtracting x + 6 the 2 square roots.

Example 4: 4 x 2 – 24 x + 36 • Are a(4 ) and 36 c( ______ Yes ) perfect squares? Yes: 2 x 6 x 2 = 24 Yes • Does √a x √c x 2 = b? ______________ Subtract • Do we have a perfect square trinomial? ______ 2 x – 6 2 x - 6 2 x – 6 2 • Do we add or subtract the square roots?

Perfect Square Trinomials What are the factors? l x 2 + 6 x + 9 = (x+3)2 2 2 (x-5) l x - 10 x + 25 = 2 l x 2 + 12 x + 36 = (x+6)

Solving Quadratic Equations by Completing the Square Is x 2 - 10 x + 25 a Perfect Square Trinomial? How do you know? What are its Factors?

Creating a Perfect Square Trinomial l In the following trinomial, the constant term is missing. X 2 + 14 x + ____ Find the constant term by squaring half the coefficient of the linear term to make it a perfect square trinomial. (14/2)2 X 2 + 14 x + 49

Perfect Square Trinomials Create perfect square trinomials. l x 2 + 20 x + ___ l x 2 - 4 x + ___ l x 2 + 5 x + ___ 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 and the constant to the right side.

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.

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+4) = 36 (x+4)2 = 36

Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve

Completing the Square-Example #2 Solve the following equation by completing the square: 2 n 2 + 12 n + 10 = 0 Step 1: Move quadratic term, and linear term to 2 n 2 + 12 n = -10 left side of the equation, the constant to the right side of the equation.

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. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first. 2 n 2 + 12 n = -10 2 n 2 + 6 n = -5 n 2 + 6 n + 9 = -5 + 9

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. n 2 + 6 n + 9 = -5 + 9 (n+3) = 4 (n+3) 2 = 4

Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side Step 5: Set up the two possibilities and solve (n+3) 2 = 4 n+3 = ± 2 n = -3 + 2 = -1 and n = -3 – 2 = -5