### Completing the square by finding the constant

You remember that a perfect square trinomial is one that can be factored into the square of a binomial.  For example, the trinomial $${x^2} + 4x + 4$$ is a perfect square trinomial because

$${x^2} + 4x + 4 = {\left( {x + 2} \right)^2}$$

Sometimes it is convenient when solving problems to be able to work with perfect square trinomials.  But not all trinomials are perfect squares.  Thankfully, a technique exists to overcome this situation.  It is called completing the square.  Here’s how it works.

For the polynomial $${x^2} + bx + c$$ (note here that the coefficient on $${x^2}$$ is 1)we can complete the square by

• Find the value of $$\Large \frac{b}{2}$$
• Square this value to obtain $${\left( {\Large \frac{b}{2}} \right)^2}$$
• Set this value as c.
• You will obtain a perfect square trinomial that can be factored as $${\left( {x + \left( {\Large \frac{b}{2}} \right)} \right)^2}$$

Let’s take a look at an example.

Example:  Find the value of c that completes the square for $${r^2} + 11r + c$$.

Solution:  Here the coefficient on r (that is, the value b) is 11.  Following the steps for completing the square we find

• The value of $$\Large \frac{b}{2}$$ is $$\Large \frac{{11}}{2}$$
• Squaring this value, we obtain $$\Large \frac{{121}}{4}$$
• So $$c = \Large \frac{{121}}{4}$$

We have now created the perfect square trinomial $${r^2} + 11r + \Large \frac{{121}}{4}$$. And, according to the last bullet of our procedure, we know that the trinomial factors as

$${r^2} + 11r + \Large \frac{{121}}{4} = {\left( {r + \Large \frac{{11}}{2}} \right)^2}$$

Congratulations, you have just completed the square.  Let’s do one more example.

Example:  Find the value of c that completes the square for $${x^2} + 38x + c$$.

Solution:  We have

• The value of $$\Large \frac{b}{2}$$ is $$\Large \frac{{38}}{2} = 19$$
• Squaring this value, we obtain 361
• Then $$c = 361$$.

We have created the perfect square trinomial $${x^2} + 38x + 361$$. It factors as

$${x^2} + 38x + 361 = {\left( {x + 19} \right)^2}$$

2047 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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1493 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch below how to solve this example:

1379 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch below how to solve this example:

### Geometry

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Beginning Algebra
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