Solving Quadratic Equations

Quadratic equations, expressions containing a squared variable, appear frequently in physics, engineering, and economics. Solving them requires finding the values of the variable that make the equation true. Three common methods exist: factoring, completing the square, and the quadratic formula.

Factoring involves rewriting the quadratic expression as a product of two linear expressions. For example, x² + 5x + 6 = 0 can be factored into (x + 2)(x + 3) = 0. Setting each factor to zero yields the solutions x = -2 and x = -3. This method is efficient when the quadratic expression is easily factorable.

Completing the square transforms the equation into a perfect square trinomial. By adding and subtracting a constant term, one side becomes a squared expression, such as (x + a)². This allows us to isolate x by taking the square root of both sides. While more involved than factoring, it provides a systematic approach applicable to all quadratic equations.

The quadratic formula is a universal solution derived by completing the square on the general quadratic equation ax² + bx + c = 0. The formula states that x = (-b ± √(b² - 4ac)) / 2a. By substituting the coefficients a, b, and c, we can directly calculate the solutions, regardless of whether the equation is factorable. The discriminant, b² - 4ac, reveals the nature of the roots: positive indicates two real roots, zero indicates one real root, and negative indicates two complex roots.