PERFECT SQUARE TRINOMIALS Any trinomial of the form

PERFECT SQUARE TRINOMIALS Any trinomial of the form ax 2 + bx + c that can be factored to be a (BINOMIAL Factor) squared Sum Factors: Difference Factors: a 2 + 2 ab + b 2 = (a + b)2 a 2 - 2 ab + b 2 = (a - b)2 (1) 9 x 2 + 12 x + 4 (2) x 2 - 8 x + 16 (3) 4 x 2 - 20 x + 25 (4) x 2 + 20 x + 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: x 2 + 6 x + c EXAMPLE: x 2 - 10 x + c Middle term: 6 Middle term: -10 Divide by 2: 3 Divide by 2: -5 Squared = 9 Squared = 25 x 2 + 6 x + 9 = (x + 3)2 x 2 – 10 x + 25 = (x - 5)2

Create Perfect Square Trinomials Practice finding “c” • x 2 - 8 x + c • x 2 - 3 x + c • x 2 + 10 x + c • x 2 + 9 x + c

Continued: Practice finding “c”

STEPS for COMPLETING THE SQUARE ax 2 + bx + c = 0 Step 1: Lead coefficient of x 2 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 • x 2 + 6 x + 4 = 0 - SUBTRACT 4 • x 2 + 6 x = - 4 -Find the constant value to create a perfect square and ADD to both sides (half of 6 is 3, 3 squared is 9) • x 2 + 6 x + 9 = -4 + 9 -FACTOR perfect square trinomial • (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 1. 2. 3. 4.

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

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

3. 4.