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UNIT 2- Quadratic Relations & Equations MA 40: Algebra 2 Task #3 – Factor & Solve Quadratic Equations Common Core: HS.F.IF.8.a Name:_______________ Date:_______Period:___ Factoring Quadratics Factoring is a powerful tool used to solve many quadratic equations. Factoring changes the quadratic from addition or subtraction form to multiplication form. When the product is equal to zero, the solution is based on the zero product property. Factoring is an algebraic method for solving. Zero Product Property: If a and b are any numbers and (a)(b) = 0, then a or b or both must be equal to zero. Example: Solve the equation x(x+5) = 0 The two factors of this equation are x and x+5. The zero product property states one of these factors must equal zero. That is, x = 0 or x + 5 = 0 Solving each equation gives the solutions: x = 0 or x = -5 The solutions of an equation occur where the graph touches or crosses the x-axis. Therefore, solutions of the equation are also called zeros of the function. The zeros on the graph are the horizontal intercepts. Use your graphing calculator to graph Y1 = x(x + 5) and verify the zeros graphically that were found algebraically. (Sketch a picture of your graph and label the zeros) Any factoring technique must begin with these steps: Set the equation equal to zero. Factor out the GCF (greatest common factor) 1. Set the equations equal to zero and factor out the GCF a. x2 – 6x = 0 b. 6x2 + 14x = 30 c. 2x3 – 8x2 + 20x = -8x You may recall that you have had previous experience with factoring out the GCF and factoring quadratic equations if a = 1. The following activity will lead you through factoring when a ≠ 1 2. Factor: 6 2 − 7 − 3 a. Think of the polynomial as fitting the form 2 + + What is a? ______ What is c? ______ What is the product ac? ______ b. List all possible pairs of integers such that their product is equal to the number ac. Organize the list in a table. Make sure that your integers are chosen so that their product has the same sign, positive or negative, as the number ac from above, and make sure that you list all the possibilities. Factors of ac (-18) Sum of factors c. What is b in the quadratic polynomial given? ______ Add the integers from each pair listed in part b. Which pair adds to the value of b from your quadratic polynomial? We will refer to these integers as m and n. m = _____ n = _____ d. Rewrite the polynomial replacing bx with mx + nx [Note either m or n could be negative. The expression indicates to add the terms mx and nx including the correct sign.] e. Factor the polynomial from part 4 by grouping. f. Check your answer by performing the indicated multiplication in your factored polynomial. Did you get the original polynomial back? 3. Use the method outlined in the previous steps to factor each of the following quadratic polynomials. Is it always necessary to list all the integer pairs whose product is ac? Explain your answer. a. 6 2 + 7 − 20 b. 2 2 + 3 − 54 c. 4 2 − 11 + 6 d. 3 2 − 13 − 10 e. 8 2 + 5 − 3 f. 18 2 + 17 + 4 g. 62 − 49 + 8 4. Compare your factorization with other groups. Did everyone write their answers the same way? Explain why answers can look different and yet be equivalent. 5. If a quadratic polynomial can be factored in the form (Ax + B)(Cx + D) where A, B, C, and D are all integers, the method you have been using will lead to the answer, specifically called the correct factorization. Show that each of these cannot be factored into the form (Ax + B)(Cx + D) using integer coefficients. a. 4z2 + z – 6 = 0 b. t2 + 2t = -8 c. 3x2 – 12 = -15x 6. As mentioned in the opening sentence of this activity, the purpose of factoring is to solve the quadratic equation. Use your factorization from Problem 3 and the zero product property to solve these quadratic equations a. 2x2 + 3x -54 = 0 b. 4w2 + 6 = 11w c. 3t2 - 13t = 10 d. 2x(4x + 3) = 3 + x e. 18z2 + 21z = 4(z – 1) f. 8 – 13p = 6p(6 – p) g. 24q2 = 4q + 8 Summary : _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________