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5-3 Solving Quadratic Equations by Graphing and Factoring A trinomial (an expression with 3 terms) in standard form (ax2 +bx + c) can be factored by finding factors that multiply to equal c and add to equal b. *Always check for a GCF first. f(x) = x2 – 4x – 12 (x + 2)(x – 6) Holt Algebra 2 c = -12, factors that multiply to equal -12 b = -4, add the factors and see which one equals -4 1 and -12 1 + (-12) = -11 -1 and 12 -1 + 12 = 11 2 and -6 2 + (-6) = -4 -2 and 6 -2 + 6 = 4 3 and -4 3 + (-4) = -1 -3 and 4 -3 + 4 = 1 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of the function by factoring. f(x)= x2 – 5x – 6 x2 – 5x – 6 = 0 (x + 1)(x – 6) = 0 x+1=0 x–6=0 x = –1 or x = 6 Holt Algebra 2 Set the function equal to 0. Factor: Find factors of –6 that add to –5. Set each parenthesis equal to zero Solve each equation. 5-3 Solving Quadratic Equations by Graphing and Factoring Example 4B: Find Roots by Using Special Factors Find the roots of the equation by factoring. 4x2 = 12x + 16 4x2 – 12x – 16 = 0 4(x2 – 3x – 4) = 0 4(x – 4)(x + 1) = 0 x–4=0 x=4 Holt Algebra 2 x+1=0 x = -1 Rewrite in standard form. Factor. The GCF is 4. Factor the trinomial Set each parenthesis equal to zero Solve each equation. 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 4a Find the roots of the equation by factoring. x2 – 4x = –4 x2 – 4x + 4 = 0 (x – 2)(x – 2) = 0 x–2=0 x=2 Holt Algebra 2 x–2=0 x=2 Rewrite in standard form. Factor the perfect-square trinomial. Set each parenthesis equal to zero. Solve each equation. 5-3 Solving Quadratic Equations by Graphing and Factoring Write a quadratic function in standard form with zeros 4 and –7. x = 4 or x = –7 x – 4 = 0 or x + 7 = 0 (x – 4)(x + 7) x2 + 3x – 28 f(x) = x2 + 3x – 28 Holt Algebra 2 Write the zeros as solutions for two equations. Rewrite each equation so that it equals 0. These two equations will represent the parenthesis had you factored the function. Multiply the binomials. Name the function. 5-3 Solving Quadratic Equations by Graphing and Factoring Example 5 Continued Check Graph the function f(x) = x2 + 3x – 28 on a calculator. The graph shows the original zeros of 4 and –7. 10 –10 10 –35 Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Determine the equation of a quadratic function with a double root at x = -1 x = -1 x+1=0 x = –1 x+1=0 (x + 1)(x + 1) x2 +2x + 1 g(x) = x2 +2x + 1 Holt Algebra 2 Write the zeros as solutions for two equations. Rewrite each equation so that it equals 0. These two equations will represent the parenthesis had you factored the function. Multiply the binomials. Name the function.