5.2 Solving Quadratic Equations by Factoring
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  • 1. 5.2 Solving QuadraticEquations by Factoring
  • 2. What is Factoring? Used to write trinomials as a product of binomials. Works just like FOIL in reverse. Example: Multiply (x + 2)(x + 7) What do you notice about the 2 and 7? 2 + 7 = 9 and 2 x 7 = 14 In general: (x + m)(x + n) = ax2 + bx + c ◦ where a=1, b = m + n, and c = mn
  • 3. Factoring x2 + bx + c (x + m)(x + n) = x2 + bx + c Need to find m and n so m+n = b and mn = c First, find all factor pairs of c. Find their sums. Choose the pair whose sum equals b. Example: Factor x2 + 5x + 6Factor Pairs Sum
  • 4. Example: What if b is negative and c is positive? ◦ Choose negative factors! x2 - 7x + 10
  • 5. Example: What if c is negative? ◦ Choose one positive and one negative! x2 - 8x – 20
  • 6. Your Turn! Factor x2 - 2x – 48
  • 7. Factoring ax2 + bx + c (a  1) Need k, l, m and n, such that: ax2 + bx + c = (kx +m)(lx + n) So, kl = a and mn = c. Find factors of a and c, then check possible answers. Example: Factor 3x2 - 17x + 10
  • 8. Example: Factor 4x2 - 4x – 3
  • 9. Your Turn! Factor 3x2 + x – 10
  • 10. Factoring Special PatternsDifference of Two Squares: a2 – b2 = (a + b)(a – b)Example: Factor x2 – 9Perfect Square Trinomial: a2 + 2ab + b2 = (a + b) 2 a2 - 2ab + b2 = (a - b) 2Example: Factor x2 + 12x + 36
  • 11. Examples: Factor 4x2 – 25 Factor 9x2 + 24x +16 Factor 49x2 – 14x + 1
  • 12. Factoring Out Monomials Check if you can factor out something from each term. Example: Factor 5x2 – 20 Example: Factor 2x2 + 8x
  • 13. Your Turn! Factor 6x2 + 15x + 9
  • 14. Solving Quadratics by Factoring Certain quadratic equations can be solved by factoring. Standard form: ax2 + bx + c = 0 Zero Product Property: ◦ Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0, or B = 0.
  • 15. Example: Solve x2 + 3x – 18 = 0
  • 16. Finding Zeros The x-intercepts of a function are also called zeros. To find zeros: ◦ Factor to rewrite in intercept form. ◦ y = ax2 + bx + c  y = a(x – p)(x – q) ◦ p and q are the zeros
  • 17. Example: Find the zeros of y = x2 – x – 6