The final method of solving quadratic equations is completing the square. The method involves putting the quadratic into the form:

y = ax² + bx + c = (x+p)² +q

where p and q are numbers.

Why the letters p and q?

I don't really know. I guess other letters were already taken.

How do I complete the square?

It can be done really quickly, since:

p = b/2

q = -(p²)

Notice that q will always be a negative number. Also notice that the numbers for a and c aren't taken into account yet.

Can you show me an example?

Here's an example question:

Complete the square for x² + 2x + 4

Here's my working out and answer:

b = 2

p = b/2 = 2/2 = 1

q = -(p²) = -(1²) = -1

y = (x+1)² -1 + 4 (add 'c' back on here)

y = (x+1)² + 3

What do you do when a is a value other than 1?

You factor it out at the beginning. Here's an example:

Complete the square for: 4x² + 2x + 4

Here's my working out and answer:

4(x² + 1/2x + 1)

b = 1/2

p = b/2 = 1/4

q = -(p²) = -(1/4²) = -1/16

y = 4((x + 1/4)² -1/16 + 1)

y = 4((x + 1/4)² + 15/16)

y = 4(x + 1/4)² + 60/16

What's the point of completing the square?

Although the quadratic formula can do everything completing the square can do, completing the square can be faster. You can get the vertex co-ordinates easily, since:

- p = vertex x co-ordinate

q = vertex y co-ordinate