Solving Quadratic Equations through Factoring

Solving Quadratic Equations through Factoring

Quadratic equations are mathematical expressions that involve variables raised to the second power, along with lower-degree terms and a constant. They take the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. In this article, we'll delve into one way to solve quadratic equations: factoring.

Fundamentals of Factoring

Factoring a quadratic equation means expressing it as the product of two simpler expressions, or factors, whose product equals the original expression. For example, we might take x² + 5x + 6 and rewrite it as x(x + 5) + 6, or as (x + 6)(x + 1). The factored form is more manageable because you can solve for each individual variable in the separate parentheses.

Factoring by Grouping

Grouping is a technique that allows us to identify factors that will simplify the quadratic expression. An expression of the form ax² + bx + c = 0 can be rearranged to look like this: a(x² + mx + n) = 0, where m = b/a and n = c/a. This allows us to find two binomials, x + r and x + s, such that m = r + s. When we factor out a, we get (x + r)(x + s).

For example, to factor x² + 6x + 5, we can see that m = b/a = 6/1 = 6. By finding factors of 6 that add up to 6 (e.g., 3 and 3), we can rewrite the expression as x(x + 3) + 5(x + 3), which is the same as (x + 3)(x + 5).

Factoring by Difference of Squares

The difference of squares is a special case where one term is the square of a binomial. For example, x² - 9 can be factored as (x + 3)(x - 3). This method is straightforward because the two binomials are already present in the expression.

Complex, Imaginary, and Real Roots

Solving a factored quadratic equation gives us two possible pairs of real roots, two complex roots (each with a real and imaginary part), or a repeated real root (also known as a double root). For instance, if we factor x² + 5x + 6 as (x + 3)(x + 2), we find the real roots x = -3 and x = -2. However, if we factor x² + 4x + 5 as (x + 2 + 3i)(x + 2 - 3i), we find that the complex roots are x = -2 ± 3i.

Two Complex Roots, Real Coefficients

If the quadratic equation has two complex roots, the discriminant, b² - 4ac, will be negative. In this case, the factored form will not be exponentially simpler than the original expression, but it can still be useful in solving for the complex roots.

Summary

Factoring a quadratic equation allows us to solve it more easily by expressing it as the product of two simpler expressions, or factors. The fundamentals of factoring involve grouping, factoring by difference of squares, and recognizing complex roots. By applying these techniques, we can simplify and solve quadratic equations effectively.