Completing the square

Completing the square is a method used to rewrite a quadratic expression, from the form ax^2+bx+c to the form a(x+d)^2+e. This can be useful for solving quadratic equations, graphing quadratic functions, and understanding the properties of a quadratic graph (e.g. its turning point, otherwise known as the vertex).

Completing the square when (a=1)

When we have a nice, simple quadratic, in the form x^2+bx+c (no x^2 coefficient) we can complete use this formula to complete the square:

x^2+bx+c = (x+\frac{b}{2})^2 - (\frac{b}{2})^2 + c

This makes a lot more sense with an example.

Example: Complete the square for x^2 + 6x + 5

• Find what b and c are:
  • b = 6
  • c = 5
• Calculate \frac{b}{2}:
  • \frac{b}{2} = \frac{6}{2} = 3
• If we take (x + 3)^2 and expand it, we get:
  • (x + 3)^2 = x^2 + 6x + 9
  • This is what we want, we just have an extra +9, so take that away from the expression:
    • x^2 + 6x + 5 = (x + 3)^2 - 9 + 5
    • = (x + 3)^2 - 4

Answer: (x + 3)^2 - 4

Completing the square when a \ne 1

When the coefficient of x^2 is not equal to 1, we can still complete the square by factoring out the coefficient first from all terms.

We have an equation equal to zero, which means we can divide by any number on the left side without worrying about changing the equation (as zero divided by any number is still zero).

Our first step is to get rid of the a coefficient (it makes things more complicated) by dividing the whole equation by a. This means the equation will now be in the form x^2 + \frac{b}{a}x + \frac{c}{a} = 0.

We can now simply use the method above to complete the square - as we don’t need to worry about the a coefficient anymore.

Example: Complete the square for 2x^2 + 8x + 6

• First, divide the whole equation by 2 to get rid of the 2 coefficient:
  • x^2 + 4x + 3
• Now, we can complete the square using the method above:
  • b = 4
  • c = 3
  • \frac{b}{2} = \frac{4}{2} = 2
  • (x + 2)^2 = x^2 + 4x + 4
  • So, we need to subtract 4 from the expression:
    • x^2 + 4x + 3 = (x + 2)^2 - 4 + 3
    • = (x + 2)^2 - 1
• Finally, we need to remember that we divided the whole equation by 2 at the start, so we need to multiply the whole completed square expression by 2 to get back to the original equation:
  • 2x^2 + 8x + 6 = 2((x + 2)^2 - 1)
  • = 2(x + 2)^2 - 2

Answer: 2(x + 2)^2 - 2

Finding the turning point from the completed square form

The turning point of a(x+d)^2+e is (-d, e).

Notice that we completely ignore the a coefficient when finding the turning point. It’s not important here!

The number in the brackets tells us the x coordinate of the turning point, and the number outside the brackets tells us the y coordinate of the turning point.

We negate the number inside the brackets to find our x coordinate. The reason for this is the same as when we solve equations from their factorised form.