Example: Factor 6x2 2x 4 – – Solution: Every term is a multiple of 2. That is, 2 is a greatest common factor, so we first factor that: 6x2 2x 4 – – = 2(3x2 x 2) – – Then we factor the trinomial in parentheses: 6x2 2x 4 – – = 2(3x2 x 2) – – = 2(3x + 2)(x 1) – 4. The Difference of Squares Some quadratic binomials are the difference of two perfect squares. When this is the case, they can be factored as a difference times a sum, and the following formula applies: a2 b2 = (a b)(a + b) – – Example: Factor 9x2 4 – Solution: 9x2 = 3x.3x and 4 = 2.2, so this binomial is the difference of two squares. It factors as: 9x2 4 = (3x 2)(3x + 2) – – Example 2: Factor 3x2 48 – Solution: 3x2 and 48 are not perfect squares, but both have 3 as a common factor. Therefore we start by factoring out 3: 3x2 48 = 3(x2 16) – – Now x2 16 is the difference of squares, so it can be – factored: 3x2 48 = 3(x2 16) = 3(x 4)(x + 4) – – – 5. Solving Equations by Factoring
Example: Factor 6x2 – 2x – 4 Solution: Every term is a multiple of 2. That is, 2 is a greatest common factor, so we first factor that: 6x2 – 2x – 4 = 2(3x2 – x – 2) Then we factor the trinomial in parentheses: 6x2 – 2x – 4 = 2(3x2 – x – 2) = 2(3x + 2)(x – 1) 4. The Difference of Squares Some quadratic binomials are the difference of two perfect squares. When this is the case, they can be factored as a difference times a sum, and the following formula applies: a2 – b2 = (a – b)(a + b) Example: Factor 9x2 – 4 Solution: 9x2 = 3x.3x and 4 = 2.2, so this binomial is the difference of two squares. It factors as: 9x2 – 4 = (3x – 2)(3x + 2) Example 2: Factor 3x2 – 48 Solution: 3x2 and 48 are not perfect squares, but both have 3 as a common factor. Therefore we start by factoring out 3: 3x2 – 48 = 3(x2 – 16) Now x2 – 16 is the difference of squares, so it can be factored: 3x2 – 48 = 3(x2 – 16) = 3(x – 4)(x + 4) 5. Solving Equations by Factoring
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