What is completing the square?
“Completing the square” is something that can be done to a quadratic expression (to make it easier to work with or more useful in some way)
It involves writing the quadratic expression x2 + px +q in the form (x + a)2 + b
Why use completing the square?
- The question might ask you to…
- You can use it to find the turning point or vertex of a quadratic curve (Parabola) – the turning point is at (−a, b)
- It’s a way of solving a quadratic equation
How to complete the square?
There are a number of slightly different ways of approaching this.
One simple process is:
- HALVE the coefficient of x to find a: a = ½p
- SUBTRACT the square of a from q to find b: b = q – a2
- WRITE DOWN the answer using your values for a and b: (x + a)2 + b
Special case: If there’s a number in front of x2, m, say, ie. m≠1
- Divide by m
- Do steps 1, 2 and 3 above
- Multiply by m