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Algebra Date _________ Chapter 8 Quadratic Expressions and Equations 8-9 Perfect Squares A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has solution Vocabulary Perfect square trinomial – Factoring Perfect Square Trinomials a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2 a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2 Example: x 2 + 8x + 16 = (x + 4)(x + 4) or (x + 4)2 X2 – 6x + 9 = (x – 3)(x – 3) = (x – 3)2 Example 1 Recognize and Factor Perfect Square Trinomials Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. a) 4y2 + 12y + 9 b) 9x2 – 6x + 4 Factoring Methods Steps Factor out the GCF Check for a difference of squares or a perfect square trinomial Apply the factoring patterns for x2+bx+c or ax2+bx+c or factor by grouping # of terms any 2 or 3 3 or 4 Examples 4x3 + 2x2 – 6x = 2x(2x2+x-3) 9x2-16=(3x+4)(3x-4) 16x2+24x+9 = (4x+3)2 X2-8x+12=(x – 2)(x _ 6) 2x2+13x+6 = (2x + 1)(x + 6) 2 12y + 9y + 8y + 6 = (12y2+9y)+(8y+6) =3y(4y+3)+2(4y+3) =(4y+3)(3y+2) Example 2 Factor Completely Factor each polynomial, if possible. If the polynomial cannot be factored write prime. a) 5x2 – 80 b) 9x2 – 6x – 35 Example 3 Solve Equations with Repeated Factors Solve 9x2 – 48x = -64 Square Root Property To solve a quadratic equation in the form x2 = n, take the square root of each side For any number n 0, if x2 = n, then x = n Example x2 = 25; x = 25 ; x = 5 Example 4 Use the Square Root Property Solve each equation. Check your solutions. a) (y – 6)2 = 81 b) (x + 6)2 = 12 Example 5 Solve an Equation During an experiment, a ball is dropped from a height of 205 feet. The formula h = 16t2+h0 can be used to approximate the number of seconds t it takes for the ball to reach height h from an initial height of h0 in feet. Find the time it takes the ball to reach the ground.