How to Solve a Quadratic Equation

How do I solve a quadratic equation step by step?

A quadratic equation is an equation in the form ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. You can solve it by factoring, completing the square, or using the quadratic formula, with the quadratic formula being the most reliable method that works for all quadratic equations.

Steps

  1. Write your equation in standard form ax² + bx + c = 0 by moving all terms to one side so the equation equals zero.
  2. Identify the values of a, b, and c from your equation, being careful with positive and negative signs.
  3. Substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
  4. Calculate the discriminant b² - 4ac first to see if you'll get real solutions, then evaluate the rest of the formula.
  5. Simplify to find both solutions, one using the plus sign and one using the minus sign in the ± symbol.

Worked example

Solve x² + 5x + 6 = 0. Here a = 1, b = 5, c = 6. Using the formula: x = (-5 ± √(25 - 24)) / 2 = (-5 ± 1) / 2. This gives x = -2 or x = -3. You can check: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0, which confirms the solution.

Remember

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) will always work for any quadratic equation, making it your most powerful tool when factoring seems difficult.

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