Understanding the Factoring Method
The factoring method solves quadratic equations of the form ax² + bx + c = 0 by factoring the expression into two binomials. Set each factor equal to zero and solve for x. This works best when the quadratic has integer factors and a leading coefficient of 1.
Key Steps in Factoring
First, ensure the equation is in standard form. Move all terms to one side and set to zero. Find two numbers that multiply to a*c and add to b. Rewrite the middle term using these numbers, then group and factor. For non-monic quadratics, factor out the greatest common divisor first.
Practical Example
Consider x² + 5x + 6 = 0. The numbers 2 and 3 multiply to 6 and add to 5, so rewrite as x² + 2x + 3x + 6 = 0. Group: x(x + 2) + 3(x + 2) = 0. Factor: (x + 2)(x + 3) = 0. Solutions: x = -2 or x = -3.
Applications and Importance
Factoring is essential in algebra for simplifying expressions and finding roots, applied in physics for projectile motion and engineering for optimization. It builds foundational skills for advanced math, though not all quadratics factor easily over integers—use the quadratic formula then.