Derive Quadratic Formula

The quadratic formula returns the x-intercept points of a quadratic equation.
Figure 1: Quadratic Formula

This example demonstrates how to derive the quadratic formula using algebra. The quadratic formula returns the two -intercepts of a quadratic equation.


  1. Start with the quadratic equation set equal to zero.

  2. Divide both sides by .

  3. Subtract from both sides.

  4. Add to both sides in order to complete the square. This prepares us to factor the left-hand side of the equation and isolate .

  5. Factor the left-side and rearrange the right-side.

  6. Combine the fractions on the right side by transforming them to have the same denominator.

  7. Take the square root of both sides.

  8. Subtract from both sides and simplify the square root expression on the right side.

  9. Combine the fractions on the right side and indicate that we are interested in both the positive and negative solution of the square root operator by using the (plus-minus) symbol.

    This completes the derivation of the quadratic formula.