5.2 Solving Quadratic Equations by Factoring Goals p Factor quadratic expressions and solve quadratic equations by factoring. p Find zeros of quadratic functions. Your Notes VOCABULARY Binomial An expression with two terms, such as x ϩ 1 Trinomial An expression with three terms, such as x2 ϩ x ϩ 1 Factoring A process used to write a polynomial as a product of other polynomials having equal or lesser degree Monomial An expression with one term, such as 3x Quadratic equation in one variable An equation that can be written in the form ax2 ϩ bx ϩ c ϭ 0 where a 0 Zero of a function A number k is a zero of a function f if f(k) ϭ 0. Example 1 Factoring a Trinomial of the Form x2 ؉ bx ؉ c Factor x2 Ϫ 9x ϩ 14. Solution You want x2 Ϫ 9x ϩ 14 ϭ (x ϩ m)(x ϩ n) where mn ϭ 14 and m ϩ n ϭ Ϫ9 . Factors of 14 1, Ϫ1, Ϫ 2, Ϫ2, Ϫ (m, n) 14 14 7 7 Sum of factors Ϫ Ϫ m ؉ n) 15 15 9 9) The table shows that m ϭ Ϫ2 and n ϭ Ϫ7 . So, x2 Ϫ 9x ϩ 14 ϭ ( x Ϫ 2 )( x Ϫ 7 ). Lesson 5.2 • Algebra 2 Notetaking Guide 97 Your Notes Example 2 Factoring a Trinomial of the Form ax2 ؉ bx ؉ c Factor 2x2 ϩ 13x ϩ 6. Solution You want 2x2 ϩ 13x ϩ 6 ϭ (kx ϩ m)(lx ϩ n) where k and l are factors of 2 and m and n are ( positive ) factors of 6 . Check possible factorizations by multiplying. (2x ϩ 3)(x ϩ 2) ϭ 2x2 ϩ 7x ϩ 6 (2x ϩ 2)(x ϩ 3) ϭ 2x2 ϩ 8x ϩ 6 (2x ϩ 6)(x ϩ 1) ϭ 2x2 ϩ 8x ϩ 6 (2x ϩ 1)(x ϩ 6) ϭ 2x2 ϩ 13x ϩ 6 The correct factorization is 2x2 ϩ 13x ϩ 6 ϭ (2x ϩ 1)(x ϩ 6) . SPECIAL FACTORING PATTERNS Difference of Two Squares Example a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b) x2 Ϫ 9 ϭ ( x ϩ 3 )( x Ϫ 3 ) Perfect Square Trinomial Example a2 ϩ 2ab ϩ b2 ϭ (a ϩ b)2 x2 ϩ 12x ϩ 36 ϭ ( x ϩ 6 )2 a2 Ϫ 2ab ϩ b2 ϭ (a Ϫ b)2 x2 Ϫ 8x ϩ 16 ϭ ( x Ϫ 4 )2 Example 3 Factoring with Special Patterns Factor the quadratic expression. a. 9x2 Ϫ 16 ϭ ( 3x )2 Ϫ 4 2 Difference of two ϭ ( 3x ϩ 4 )( 3x Ϫ 4 ) squares b. 16y2 ϩ 40y ϩ 25 Perfect square trinomial ϭ ( 4y )2 ϩ 2( 4y )( 5 ) ϩ 5 2 ϭ ( 4y ϩ 5 )2 c. 64x2 Ϫ 32x ϩ 4 Perfect square trinomial ϭ ( 8x )2 Ϫ 2( 8x )( 2 ) ϩ 2 2 ϭ ( 8x Ϫ 2 )2 98 Algebra 2 Notetaking Guide • Chapter 5 Your Notes Example 4 Factoring Monomials First Factor the quadratic expression. a. 12x2 Ϫ 3 ϭ 3 (4x2 Ϫ 1) ϭ 3 (2x Ϫ 1)(2x ϩ 1) b. 3u2 Ϫ 9u ϩ 6 ϭ 3 (u2 Ϫ 3u ϩ 2) ϭ 3 (u Ϫ 2)(u Ϫ 1) c. 7v2 Ϫ 42v ϭ 7v (v Ϫ 6) d. 2x2 ϩ 8x ϩ 2 ϭ 2 (x2 ϩ 4x ϩ 1) Checkpoint Factor the expression. 1. 6c2 Ϫ 48c Ϫ 54 2. 81x2 Ϫ 1 6(c ϩ 1)(c Ϫ 9) (9x ϩ 1)(9x Ϫ 1) 3. 49h2 ϩ 42h ϩ 9 4. 16x2 Ϫ 4 (7h ϩ 3)2 4(2x ϩ 1)(2x Ϫ 1) ZERO PRODUCT PROPERTY Let A and B be real numbers or algebraic expressions. If AB ϭ 0, then A ϭ 0 or B ϭ 0 . Example 5 Solving Quadratic Equations Solve 4x2 ϩ 13x ϩ 11 ϭϪ3x Ϫ 5. Solution 4x2 ϩ 13x ϩ 11 ϭϪ3x Ϫ 5 Write original equation. 4x2 ϩ 16x ϩ 16 ϭ 0 Write in standard form. x2 ϩ 4x ϩ 4 ϭ 0 Divide each side by 4 . (x ϩ 2)2 ϭ 0 Factor. x ϩ 2 ϭ 0 Use zero product property. x ϭ Ϫ2 Solve for x. The solution is Ϫ2 . Check this in the original equation. Lesson 5.2 • Algebra 2 Notetaking Guide 99 Your Notes Checkpoint Solve the quadratic equation. 5. x2 ϩ 15x ϩ 26 ϭ 0 6. 2x2 ϩ x ϩ 3 ϭ Ϫ5x ϩ 19 ϩ x2 Ϫ13, Ϫ2 Ϫ8, 2 Example 6 Finding the Zeros of a Quadratic Function .x2 ؉ 4x ؉ 3 ؍ Find the zeros of y Use factoring to write the function in intercept form. y ϭ x2 ϩ 4x ϩ 3 ϭ (x ϩ 3)(x ϩ 1) The zeros of the function are Ϫ3 and Ϫ1 . Check Graph y ϭ x2 ϩ 4x ϩ 3. The graph passes through ( Ϫ3 , 0) and ( Ϫ1 , 0), so the zeros are Ϫ3 and Ϫ1 . Checkpoint Complete the following exercise. 7. Find the zeros of y ϭ 3x2 Ϫ x Ϫ 2. Homework 2 Ϫᎏᎏ, 1 3 100 Algebra 2 Notetaking Guide • Chapter 5.