Solving quadratic equations
Solving a quadratic equation means finding the values of x that satisfy ax² + bx + c = 0. The four standard methods are factoring, taking square roots, completing the square, and applying the quadratic formula.
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a ≠ 0. Its solutions — also called roots or zeros — are the x-values where the corresponding parabola crosses the x-axis. A quadratic has at most two real solutions.
Choose the method that fits the equation. Factoring is fastest when the expression breaks apart cleanly: x² − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3, because a product equals zero only when one factor equals zero. Taking square roots works when there is no linear term: x² = 49 gives x = ±7. Completing the square rewrites the equation with a perfect-square trinomial and also converts it to vertex form. The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), always works, factorable or not.
The expression under the radical, b² − 4ac, is the discriminant, and it tells you what to expect before you finish solving. A positive discriminant means two distinct real solutions, a discriminant of zero means one repeated real solution where the parabola touches the axis at its vertex, and a negative discriminant means no real solutions — the roots are complex and the parabola never crosses the x-axis.
Always rearrange to standard form first: an equation like x² = 5x − 6 must become x² − 5x + 6 = 0 before factoring or applying the formula. The ACT tests quadratics in its intermediate algebra section, including factorization and solving, while the FE Mechanical exam covers quadratics under analytic geometry and supplies the quadratic formula in its reference handbook.
Key takeaways
- Put the equation in standard form ax² + bx + c = 0 before solving.
- Factoring, the square root property, completing the square, and the quadratic formula are the four standard methods.
- The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) solves any quadratic, including ones that will not factor.
- The discriminant b² − 4ac tells you whether there are two real roots, one repeated root, or no real roots.