Solving quadratic equations by completing the square

We can solve quadratic equations by completing the square.

Steps

1. Make the quadratic in the form if it isn’t already.
2. If , divide the whole equation by to get rid of the coefficient of .
3. Move the constant term to the other side of the equation.
4. Complete the square on the left side of the equation using the method from the last page.
5. Move any constants from the left side to the right side by adding or subtracting.
6. Take the square root of both sides (remembering to include both the positive and negative roots).
7. Solve for by moving any constants to the right side.

Examples

Example: Solve by completing the square

• Move the constant to the other side:
• Complete the square on the left side:
• Take the square root of both sides:
• Solve for :
  • or

Answer: or

Example: Solve by completing the square

• Divide the whole equation by :
• Move the constant to the other side:
• Complete the square on the left side:
• Take the square root of both sides:
• Solve for :
  • or

Answer: or