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8-3A Factoring Trinomials and Solving Quadratic Equations There are numerous methods to factor trinomials. The method used in this presentation is NOT in your textbook. Please pay attention as this method is easier to use than the method presented in the book! Algebra 1 Glencoe McGraw-Hill Linda Stamper In the previous lesson, you solved a quadratic equation by factoring. The problem. Set each factor equal to zero and solve! x 3x 5 0 x 3 0 x 3 or or x5 0 x 5 The factors were given information. Today you will need to find the factors of a quadratic trinomial and then use the factors to solve a quadratic equation. Quadratic expressions are written in the following way: ax2 bx c quadratic trinomial leading coefficient Today we will factor trinomials when the leading coefficient is 1. x 2 bx c The coefficient of x 2 is 1. When the coefficient of ax 2 is 1. ax2 bx c becomes x 2 bx c ac Product ac b Sum b (a = 1). To factor the trinomial we will. Multiply a times c and place this on top. And place b in the bottom. To fill in the sides of the x you must find two numbers that have a product of ac and a sum of b. Factor. Multiply a times c to find the product. ax2 bx c 15 x2 8x 8 15 5 3 Draw an X on your x x b in the bottom represents the sum paper. To fill in the sides of the x you must find two This quadratic numbers that have a product of trinomial 15 and a sum of 8. is an expression. Place the values from the sides of theHow X figure into do you know it is NOT your factors. and equation? Check by doing You know FOIL in your there will be head! an x in each factor! All of today’s problems involving quadratic trinomials will have a leading coefficient of 1. Multiply a times c to find the product. –3 ax2 bx c x2 –22x 3 –33 1 x x b in the bottom represents the sum To fill in the sides of the x you must find two numbers that have a product of –3 and a sum of –2. Place the values from the sides of the X figure into your factors. Check by doing You know FOIL in your there will head! be an x in each factor! Example 1 Factor. 1. Write the problem. 2. Draw an X next to the problem. ax2 bx c x2 7x 12 x x 12 +4 4 +3 3 7 3. Multiply a times c to find the product. 4. Write b in the bottom to represent the sum. 5. Fill in the sides of the x by finding two numbers that have a product of the top Check by number and a sum of the bottom number. doing FOIL 6. Using the values from the sides of in your the X figure write the factors. Your head! factors must be in parentheses! Solve. 1) x 2 4 x 3 0 x 1x 3 0 x + 1 = 0 or x + 3 = 0 - 1 -1 or - 3 -3 x = -1 or x=-3 -1 and -3 are the x-intercepts (roots, zeros or solutions of the quadratic function: y x2 4x 3 Recall that to find the x-intercepts, you let y = 0. You do the same for a quadratic function. This will graph a PARABOLA! Axis of symmetry x= = -4 = -2 2(1) Quadratic Function y = ax2 + bx + c y x2 4x 3 Let x = 0 y-intercept ( 0, c ) (0,3) Quadratic Equation ax2 + bx + c = 0 x2 4x 3 0 Vertex ( x-intercept’s -1, -3 , y) Let y = 0 Factor and solve x2 4x 3 0 (x +1) (x + 3) = 0 (-2, -1) a= 1 b= 4 c= 3 a>0 so parabola opens up Vertex : at the vertex, x=-2 Substitute x=-2 into the quadratic function y x2 4x 3 Vertex = (-2, -1) y = (-2)2 + 4(-2) + 3 = -1 Vertex = (-2, -1) 8-A6 Page 438 # 12–31.