The quadratic formula solves quadratic equations written in the standard form ax² + bx + c = 0, where a is not zero. It is especially useful when an equation does not factor neatly into simple expressions. The formula provides both solutions, including repeated or complex roots when applicable.

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

The expression b² − 4ac is called the discriminant. It helps determine the type of solutions before completing the calculation.

Worked example

Solve x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1. Substituting gives x = (5 ± 1) ÷ 2, so the two roots are x = 3 and x = 2.

How to use the formula

  1. Rewrite the equation in standard form.
  2. Identify a, b, and c, including their signs.
  3. Calculate the discriminant b² − 4ac.
  4. Substitute the values into the formula.
  5. Simplify both the plus and minus cases.
  6. Check each solution in the original equation.

Understanding the discriminant

A positive discriminant gives two distinct real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two complex conjugate roots, so the square root involves the imaginary unit i.

Common errors

Pay particular attention to negative signs and brackets when substituting b. The denominator is 2a, not just 2. The method also requires a ≠ 0; if a is zero, the equation is not quadratic.

Frequently asked questions

Does the quadratic formula always work?

It applies to quadratic equations with real or complex coefficients in standard form, with a not equal to zero.

Why are there two signs?

The plus and minus operations produce the two possible roots.

How can I check an answer?

Substitute each root into the original equation and verify that the left side equals zero, allowing for rounding if decimals are used.