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how to complete the square?

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Completing the Square

"Completing the Square" is where we ...

take a Quadratic Equation
like this:and turn it
into this:ax2 + bx + c = 0a(x+d)2 + e = 0

For those of you in a hurry, I can tell you that:d = b2a

and:e = c − b24a

Completing the Square

Say we have a simple expression like x2 + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x2 + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)2

In Algebra it looks like this:

x2 + bx+ (b/2)2=(x+b/2)2
"Complete the

So, by adding (b/2)2 we can complete the square.

The result of (x+b/2)2 has x only once, which is easier to use.

Keeping the Balance

Now ... we can't just add (b/2)2 without also subtracting it too! Otherwise the whole value changes.

So let's see how to do it properly with an example:

Start with:

("b" is 6 in this case)

A Shortcut Approach

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 we get: x2 + 2dx + d2 + e

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

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 x2:

ax2 + bx + c = 0

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

x2 + (b/a)x + c/a = 0


Now we can solve a Quadratic Equation in 5 steps:

  • Step 1 Divide all terms by a (the coefficient of x2).
  • Step 2 Move the number term (c/a) to the right side of the equation.
  • Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p)2 = q, which can be solved rather easily:

  • Step 4 Take the square root on both sides of the equation.
  • Step 5 Subtract the number that remains on the left side of the equation to find x.

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