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Completing the Square (Shortcut)

Completing the Square is where we take a Quadratic Equation

like this ax^2+ bx + c = 0 and turn it into this a(x + d)^2 + e = 0

For those of you in a hurry, I can tell you that:

d = b / 2a

e = c − [ (b^2) / 4a ]

Too short for you to even understand? Let's look at another shortcut but not so short method.

Here is a quick way to get an answer. You may like this method.

First think about the result we want: (x+d)^2 + e

After expanding (x+d)^2 + e we get: x^2 + 2dx + d^2 + e

Now see if we can turn our example into that form to discover d and e


try to fit x^2 + 6x + 7 into x^2 + 2dx + d^2 + e

Now we can "force" an answer:

  • We know that 6x must end up as 2dx, so d must be 3

  • Next, we see that 7 must become d^2 + e = 9 + e, so e must be −2

And we get the answer (x+3)^2 − 2

Solving General Quadratic Equations by Completing the Square

We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have a coefficient of a in front of x^2:

ax^2 + bx + c = 0

But that is easy to deal with ... just divide the whole equation by "a" first, then carry on: