What is the Factoring Method for Quadratic Equations?
The factoring method solves quadratic equations of the form ax² + bx + c = 0 by factoring the quadratic expression into two binomials. Set each binomial equal to zero and solve for x. This works when the quadratic factors nicely over the integers, providing exact roots without decimals.
Key Steps to Factor and Solve
First, ensure the equation is in standard form. For ax² + bx + c = 0, find two numbers that multiply to a*c and add to b. Rewrite the middle term using these numbers, group, and factor by grouping. Alternatively, for a=1, directly find factors of c that add to b. Set (x + m)(x + n) = 0, so x = -m or x = -n.
Practical Example: Solving x² + 5x + 6 = 0
Consider x² + 5x + 6 = 0. The numbers 2 and 3 multiply to 6 and add to 5, so it factors as (x + 2)(x + 3) = 0. Setting each factor to zero gives x + 2 = 0 (x = -2) and x + 3 = 0 (x = -3). Verify by plugging back: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0, and similarly for -3.
Applications and Importance of Factoring
Factoring is essential in algebra for finding roots, which model real-world scenarios like projectile motion or profit maximization. It's faster than the quadratic formula for simple cases and builds foundational skills for advanced math, though it requires practice to spot factorable quadratics quickly.