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Completing the Square

Written by tutor Susan L.

Completing the square may seem a bit odd, at first, since the easiest way to learn it is to use it to solve quadratic equations. You remember quadratic equations -- those that look like this:

Sure, there are other methods that you can use to solve basic quadratic equations (like factoring or the quadratic formula); however, completing the square is a good tool that you will need when working on other types of equations – like parabolas, eclipses, circles, and hyperbolas.

Follow along with this example:

Solve the quadratic equation:

Now, you can see that solving this by factoring is a bit impossible.

You could certainly use the quadratic formula to solve it, but you could also use ‘completing the square’ to find the solution – as shown below:



First you must prepare the equation (using steps 1 – 3).

STEP 1:

First, move the ‘ones’ number or ‘loose number’ over to the other side of the equals sign
by subtracting 9 from BOTH sides.

STEP 2:

Next, we want the coefficient of the squared term to be ONE. So, we factor a 3 out of the left terms.

STEP 3:

Then we divide BOTH sides by 3


Now we are ready to ‘complete the square’

STEP 4:

Take the coefficient of the ‘non squared’ x term (in this case 2), divide it by 2 and then square it
-->
so we ADD ONE to BOTH sides of the equation

STEP 5:

Now we can factor the left side, and combine like terms on the right.

STEP 6:

Since we are solving for x, we need to take the square root of BOTH sides.


Remember, the result of taking a square root leads to both a positive and negative result.

STEP 7:

Now we subtract 1 from both sides, in order to get x by itself on the left side.

The FINAL SOLUTION:

The solution(s) to this quadratic equation are the following:

AND


To be mathematically proper, a solution should be shown in terms of i if there are negative values beneath the square root sign.


So, the properly written solution(s) to this quadratic equation are the following:

AND

Why choose complete the square?

Why would you choose to ‘complete the square’ rather than using the other methods you already know?

Well, completing the square really comes in handy when converting parabolic, hyperbolic, circular, etc. equations to their standard form. The reasoning behind converting to ‘standard form’ has to do with gathering very interesting information directly from the equation.

For example, the standard form of a parabolic equation looks like this:

This form of the equation provides us with the vertex of the parabola, which is found at the point (h, k).

Starting with a typical hyperbolic equation: ,
you can see that we cannot easily determine the vertex from this equation.

But, if we convert this into the standard form, we can easily see this information. So we will convert this equation by ‘completing the square.’

STEP 1:

First, we want to move the ‘ones’ number or loose number to the right of the equals sign by subtracting 24 from both sides.

We really want all the x’s on the left side of the equation – so we also add 3y to BOTH sides.

STEP 2:

Complete the square.
Take the non-squared x-term and divide by 2 – then square that value.
6/2 = 3 --> 32 = 9 --> so we ADD NINE to BOTH sides of the equation

STEP 3:

Now we can factor the left side, and combine like terms on the right side.

STEP 4:

Remember, we want this to look like the standard form:
So we want to also factor a 3 out of the right side.

Now, we have the standard form of this equation, and we can see that h = -3 and k = 5.

This provides us with the vertex of the parabola, which is at the point (h, k) or (-3, 5) in this case.

Completing the Square Quiz

Question 1. In order to ‘complete the square’, what do you do to the ‘middle’ or non-squared x term? (choose 1)
(i.e. x2 + 6x = 15)

A. Multiply by 2 and then square the result
B. Divide by 2 and then square the result
C. Multiply by 2 and then add one to the result
The correct answer here would be B.

Question 2. What do you do with the result from Question #1? (choose 1)

A. Subtract it from the left side
B. Add it to the left side
C. Subtract it from BOTH sides
D. Add it to BOTH sides
E. Subtract it from the right side
F. Add it to the right side
The correct answer here would be D.

Question 3. When CAN'T you use ‘completing the square’ ? (choose 1)

A. To solve a quadratic equation
B. To convert the equation of a circle to standard form
C. To find the vertex of a parabola
D. To find the intersection of two lines
E. To avoid using the quadratic equation
The correct answer here would be D.
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