Completing the square

Completing the square enables us to change an expression of the form

ax² + bx + c

to the form

a(x + d)² + e.

The reason that completing the square is useful here is that we can always solve an equation of the form

a(x + d)² + e = 0

by

1. subtracting e from both sides
2. dividing by a
3. taking the square root
4. subtracting d.

Now the question is, how do we complete the square?

We know that

(x + d)² = x² + 2dx + d²

To get the constant term, you take the coefficient of x, halve it, and then square it.

Now do this to x² + bx + c.

Halving the coefficient of x gives .

Now square this to get .

So

.

But we don't have that, we have x² + bx + c.

However,

This is what it means to complete the square for x² + bx + c.

We can also complete the square for ax² + bx + c.

We can do this by

1. dividing through by a
2. completing the square
3. multiplying by a.