Factoring Special Cases Factor completely: 9x2 + 66x + 21

Difference of Perfect Squares For all numbers a and b, a2 – b2 = (a-b) (a+b) Example 1: (x2 - 25) Since 25 is a perfect square: a2 = x2 and b2 = 25 Therefore: a = ___ and b = ___ Solution: (x2 - 25) = ( ) ( ) x 5 x - 5 x + 5

Example 2: 10x2 - 40 10 (x2 – 4) First factor out the 10: __________ Now we have a difference of perfect squares ( ) Note that it has to be a difference so the inner and outer products can cancel each other out. Solution: ( ) ( ) x2 and 4 10 x-2 x+2

Perfect Square Trinomials Example 3: x2 + 12x + 36 (ax2 + bx + c) To identify a perfect square trinomial, See if a (1) and c (36) are perfect squares. Multiply square roots of a & c & 2. ________________ If the answer equals b (12), you have a perfect square trinomial. __________________________________ Factor by adding/subtracting the square roots. Solution: x2 +12x +36 = ( ) ( ) = ( ) 1 x 6 x 2 = 12 x + 6 x + 6 x + 6 2

Example 4: 4x2 – 24x + 36 Yes 2x – 6 2x - 6 2x – 6 2 Are a( ) and c( ) perfect squares? ____________ Does √a x √c x 2 = b? ___________________________ Do we have a perfect square trinomial? ____________ Do we add or subtract the square roots? ____________ Solution: 4x2 – 24x + 36 = ( ) ( ) = ( ) Yes 4 36 Yes: 2 x 6 x 2 = 24 Yes Subtract 2x – 6 2x - 6 2x – 6 2

Perfect Square Trinomials What are the factors? x2 + 6x + 9 = x2 - 10x + 25 = x2 + 12x + 36 = (x+3)2 (x-5)2 (x+6)2

Solving Quadratic Equations by Completing the Square Is x2 - 10x + 25 a Perfect Square Trinomial? How do you know? What are its Factors?

Creating a Perfect Square Trinomial In the following trinomial, the constant term is missing. X2 + 14x + ____ Find the constant term by squaring half the coefficient of the linear term to make it a perfect square trinomial. (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 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) (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: Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation. 2n2 + 12n + 10 = 0 2n2 + 12n = -10

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. 2n2 + 12n = -10 2 n2 + 6n = -5 n2 + 6n + 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. n2 + 6n + 9 = -5 + 9 (n+3) (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 n = -3 + 2 = -1 and n = -3 – 2 = -5