Derive Quadratic Formula

This example derives the quadratic formula using algebra.


  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 to complete the square which prepares us to factor the left-hand side of the equation and isolate .

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

  6. Multiply the expression by one in the form of so the two fractions on the right side can be combined.

  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.