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4.4 Solve Quadratic Equations of the form b y Factoring

4.4 Solve Quadratic Equations of the form b y Factoring. EXAMPLE 1. Factor ax 2 + bx + c where c > 0. Factor 5 x 2 – 17 x + 6. SOLUTION.

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4.4 Solve Quadratic Equations of the form b y Factoring

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  1. 4.4 Solve Quadratic Equations of the form by Factoring

  2. EXAMPLE 1 Factor ax2 + bx + c where c > 0 Factor 5x2 – 17x + 6. SOLUTION You want 5x2 – 17x + 6 = (kx + m)(lx + n) where kand lare factors of 5 andmand nare factors of 6. You can assume that kand lare positive and k ≥ l. Because mn> 0, mand nhave the same sign. So, mand nmust both be negative because the coefficient of x, –17, is negative.

  3. ANSWER The correct factorization is5x2 –17x + 6 = (5x – 2)(x – 3). EXAMPLE 1 Factor ax2 + bx + c where c > 0

  4. ANSWER The correct factorization is3x2 + 20x – 7= (3x – 1)(x + 7). EXAMPLE 2 Factor ax2 +bx + c where c < 0 Factor 3x2 + 20x – 7. SOLUTION You want3x2 + 20x – 7 = (kx + m)(lx + n)wherekandlarefactors of3andmandnare factors of–7. Becausemn < 0, mandn have opposite signs.

  5. for Examples 1 and 2 GUIDED PRACTICE GUIDED PRACTICE Factor the expression. If the expression cannot be factored, say so. 1. 7x2 – 20x – 3 ANSWER (7x + 1)(x – 3) 2. 5z2 + 16z + 3 ANSWER (5z + 1)(z + 3). 3. 2w2 + w + 3 ANSWER cannot be factored

  6. for Examples 1 and 2 GUIDED PRACTICE GUIDED PRACTICE 4. 3x2 + 5x – 12 ANSWER (3x – 4)(x + 3) 5. 4u2 + 12u + 5 ANSWER (2u + 1)(2u + 5) 6. 4x2 – 9x + 2 ANSWER (4x – 1)(x –2)

  7. EXAMPLE 3 Factor with special patterns Factor the expression. a. 9x2 – 64 = (3x)2 – 82 Difference of two squares = (3x + 8)(3x – 8) b. 4y2 + 20y + 25 = (2y)2 + 2(2y)(5) + 52 Perfect square trinomial = (2y + 5)2 c. 36w2 – 12w + 1 = (6w)2 – 2(6w)(1) + (1)2 Perfect square trinomial = (6w – 1)2

  8. for Example 3 GUIDED PRACTICE GUIDED PRACTICE Factor the expression. 7. 16x2 – 1 (4x + 1)(4x – 1) ANSWER 8. 9y2 + 12y + 4 (3y + 2)2 ANSWER 9. 4r2 – 28r + 49 (2r – 7)2 ANSWER 10. 25s2 – 80s + 64 ANSWER (5s – 8)2

  9. for Example 3 GUIDED PRACTICE GUIDED PRACTICE 11. 49z2 + 4z + 9 ANSWER (7z + 3)2 12. 36n2 – 9 = (3y)2 ANSWER (6n – 3)(6n +3)

  10. EXAMPLE 4 Factor out monomials first Factor the expression. = 5(x2 – 9) a. 5x2 – 45 = 5(x + 3)(x – 3) b. 6q2 – 14q + 8 = 2(3q2 – 7q + 4) = 2(3q – 4)(q – 1) c. –5z2 + 20z = –5z(z – 4) d. 12p2 – 21p + 3 = 3(4p2 – 7p + 1)

  11. for Example 4 GUIDED PRACTICE GUIDED PRACTICE Factor the expression. 13. 3s2 – 24 ANSWER 3(s2 – 8) 14. 8t2 + 38t – 10 ANSWER 2(4t – 1) (t + 5) 15. 6x2 + 24x + 15 ANSWER 3(2x2 + 8x + 5) 16. 12x2 – 28x – 24 ANSWER 4(3x + 2)(x – 3) 17. –16n2 + 12n ANSWER –4n(4n – 3)

  12. for Example 4 GUIDED PRACTICE GUIDED PRACTICE 18. 6z2 + 33z + 36 ANSWER 3(2z + 3)(z + 4)

  13. orx + 4 = 0 3x – 2 = 0 x = orx = –4 23 EXAMPLE 5 Solve quadratic equations Solve(a) 3x2 + 10x – 8 = 0 and (b) 5p2 – 16p + 15 = 4p – 5. a. 3x2 + 10x – 8 = 0 Write original equation. (3x – 2)(x + 4) = 0 Factor. Zero product property Solve for x.

  14. EXAMPLE 5 Solve quadratic equations b. 5p2 – 16p + 15 = 4p – 5. Write original equation. 5p2 – 20p + 20 = 0 Write in standard form. p2 – 4p + 4 = 0 Divide each side by 5. (p – 2)2 = 0 Factor. p – 2 = 0 Zero product property p = 2 Solve for p.

  15. 3 or –3 12 GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 Solve the equation. 19. 6x2 – 3x – 63 = 0 ANSWER 20. 12x2 + 7x + 2 = x +8 no solution ANSWER 21. 7x2 + 70x + 175 = 0 ANSWER –5

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