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
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
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
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.
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:
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:
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: