Completing the Square

Quadratic expression

• If a quadratic expression is written in the form (x+p)^2 + q it is in completed square form
• You can solve quadratic equations which don’t have integer answers by completing the square

Useful identities

• If you learn these two identities you can save time when you are completing the square
  • x^2 + 2bx + c = (x+b)^2 - b^2 + c
  • x^2 - 2bx + c = (x-b)^2 - b^2 + c

Positive and negative roots

• Remember that any positive number has two square roots: one positive and one negative
• If you square root both sides of an equation you need to use plus-or-minus to show that there are two square roots

Working it out

• Compare the expression with the identities for completing the square
• Substitute these values into the identity and simplify to find the numbers values
• Use your answer to part a to write the expression in completed square form
• The unknown only appears once so you can solve it using inverse operations