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Park Bench Another Method of Factoring Objective: To Factor Quadratics in the form of ax2 ± bx ± c = 0, where a, b, and c are real numbers. NOTE: The method demonstrated in this handout is most effective when the leading coefficient of the quadratic equation is not equal to one. a ≠ 1 Example: Factor the following quadratic equation into its linear factors: 6 x 2 − 17 x + 12 Step 1 Find the key by multiplying the values for a and c. The general equation for a quadratic is ax 2 ± bx ± c ; therefore, in our example a = 6, b = -17 and c = 12. The key is formed by multiplying a * c, or 6 * 12 = 72 Step 2 List the factors for the key value. If the key value is negative, one of the factors must be negative. If the key is positive and the b value is negative, all factors of the key are negative. In the example, the key value is equal to 72 and the value for b is negative (b = -17); therefore, all the factors listed must be negative. Factors of 72 -1 * -72 -2 * -36 -3 * -24 -4 * -18 -6 * -12 -8 * -9 Step 3 Determine which factors of the key will give a total value equal to the value for b when added. Using the list of factors, determine what the sum of the factors would be. Look for the factors that give you a sum equal to the value of b (b = -17). Factors of 72 -1 * -72 -2 * -36 -3 * -24 -4 * -18 -6 * -12 -8 * -9 The Math Center ■ Sum of the Factors -73 -38 -27 -22 -18 -17 Valle Verde ■ Tutorial Support Services ■ EPCC 1 factor )(ax + factor ) , where a is the value of the leading a coefficient and the factors are those identified in step 3. Step 4 Use the following formula, (ax + In the example, a = 6 and the factors are -8 and -9. (ax + (6 x − 8)(6 x − 9) factor )(ax + factor ) →→ a 6 Step 5 Factor out common multiples from each of the linear factors. (6 x − 8)(6 x − 9) 6 →→ 2(3x − 4) ∗ 3(2 x − 3) 6 Step 6 The common multiples that were factored out should cancel with the denominator leaving only the linear factors. 2(3x − 4) ∗ 3(2 x − 3) 6 ∗ (3x − 4 )(2 x − 3) 6 ∗ (3x − 4 )(2 x − 3) →→ →→ → → (3 x − 4 )(2 x − 3) 6 6 6 Therefore, the linear factors of 6 x 2 − 17 x + 12 are (3 x − 4 )(2 x − 3) . Listed below are other factoring techniques. Special Factor Formulas: x3 + a3 = (x + a) (x2 – ax + a2) x3 – a3 = (x – a) (x2 + ax + a2) x2 – a2 = (x – a) (x + a) x4 – a4 = (x2 – a2) (x2 + a2) x4 + a4 = (x2 + 2 ax + a2) (x2 – 2 ax + a2) Factor by Grouping: acx3 + adx2 + bcx + bd = ax2 (cx + d) + b (cx + d) = (ax2 + b) (cx + d) The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC 2