PERFECT SQUARE TRINOMIALS Any trinomial of the form ax2 + bx + c that can be factored to be a (BINOMIAL Factor) squared Sum Factors: a2 + 2ab + b2 = (a + b)2 Difference Factors: a2 - 2ab + b2 = (a - b)2 (1) 9x2 + 12x + 4 (2) x2 - 8x + 16 (3) 4x2 - 20x + 25 (4) x2 + 20x + 100

How do you make a perfect square trinomial? STEP 1: DIVIDE middle term value (b-value) by 2 STEP 2: SQUARE it STEP 3: Make your step 2 answer the constant FACTORS: Binomial is add if middle term is positive Binomial is subtract if middle term is negative EXAMPLE: x2 + 6x + c EXAMPLE: x2 - 10x + c Middle term: 6 Middle term: -10 Divide by 2: 3 Divide by 2: -5 Squared = 9 Squared = 25 x2 + 6x + 9 = (x + 3)2 x2 – 10x + 25 = (x - 5)2

Create Perfect Square Trinomials Practice finding “c” x2 - 8x + c x2 + 10x + c x2 - 3x + c x2 + 9x + c

Continued: Practice finding “c”

STEPS for COMPLETING THE SQUARE ax2 + bx + c = 0 Step 1: Lead coefficient of x2 must be 1 DIVIDE by “a” value Step 2: Subtract current ‘c’ term Step 3: Find value to make a perfect square trinomial Divide middle term, “bx”, by 2 and square Add that value to both sides of equation Step 4: Factor (perfect square!) *Shortcut = half of middle term is part of binomial factor* Step 5: Solve for x

Example: Solve by completing the square x2 + 6x + 4 = 0 - SUBTRACT 4 x2 + 6x = - 4 Find the constant value to create a perfect square and ADD to both sides (half of 6 is 3, 3 squared is 9) -FACTOR perfect square trinomial x2 + 6x + 9 = -4 + 9 (x + 3)2 = 5 SOLVE for x: Square root both sides Use plus or minus (Check to simplify radical)

Practice #1: Completing the Square 1. 2. 3. 4.

Example with leading coefficient - Divide every number by 2 - Add 3/2 on both sides Find c to make perfect square trinomial (half of 2 = 1, 1 squares = 1 - Factor left side, combine like terms on the right - Solve for x: Square Root with plus/minus Rationalize Fraction Radicals

Practice #2: Completing the Square 2. 1. 3. 4. Math 3 Hon: Unit 3

Practice: Equations with Complex Solutions 1. 2. 3. 4.

Practice : Solve Equations to equal zero? 1. 2.

3. 4.