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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

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

Below you can download some free math worksheets and practice.


Downloads:
1750 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:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-easy

Watch below how to solve this example:

 

Downloads:
1217 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:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-medium

Watch below how to solve this example:

 

Downloads:
1105 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:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-hard

Watch below how to solve this example:

 
 
 

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