Algebra:
Completing the Square

1. 
\(3{x}^{2}+5x+4\)
 Solution
2. 
\({x}^{2}+7x\)
 Solution
3. 
\(3{x}^{2}+7x\)
 Solution

Complete the Square - Introduction

A quadratic equation is any equation in the form
\(a{x}^{2}+bx+c=0\)
, where x is the unknown, and a, b, and c are known numbers, with a ≠ 0. The numbers a, b, and c are the coefficients of the equation and are called respectively, the quadratic coefficient, the linear coefficient and the constant term.
Completing the square is the process of converting a quadratic expression from the form:
\(a{x}^{2}+bx+c\)

into the form:
\(a{(x-h)}^{2}+k\)
Why do we do this? The most common uses of completing the square are the following:
1) Solve quadratic equations.
2) Graph quadratic functions.
3) Solve integrals in calculus.

How to Complete the Square

To oomplete the square, first check your problem. If an expression follows the form
\(a{x}^{2}+bx+c\)
, it is a candidate for this technique. Remember: Expressions such as
\(2{x}^{2}+5x\)
also fit this pattern — in this case,
\(c\)
is zero.
Consider the first example below:
\(3{x}^{2}+5x+4\)
Step 1: Assign values using the template formula
\(a{x}^{2}+bx+c\)
. In this case,
\(a\)
=
\(3\)
,
\(b\)
=
\(5\)
and
\(c\)
=
\(4\)
.
Step 2: Factor out
\(a\)
, which is 3, to give
\(3({x}^{2}+\frac{5}{3}x+\frac{4}{3})\)
.
Step 3: Introduce a constant, "
\(k\)
", which is obtained by using the formula
\(k={(\frac{b}{2a})}^{2}\)
.
In our equation, that yields
\(\frac{25}{36}\)
, which must be both added and subtracted to the equation so its value does not change. This results in:
\(3({x}^{2}+\frac{25}{36}-\frac{25}{36}+\frac{4}{3})\)
Step 4: Factor using the Square of Difference Rule, which yields:
\(3({(x+\frac{5}{6})}^{2}-\frac{25}{6}+\frac{4}{3})\)
.
Step 5: Simplify by the two fractions at the end to produce:
\(3({(x+\frac{5}{6})}^{2}+\frac{23}{36})\)
.
Step 6: Finally, expand the equation to give:
\(3{(x+\frac{5}{6})}^{2}+\frac{23}{12}\)
.
We are done! The result is a simplified polynomial with only one
\(x\)
variable.

What's Next

Ready to give it a try? See if you can solve our completing the square practice problems at the top of this page, and use our step-by-step solutions if you get stuck.
At Cymath, not only do we aim to help you understand the process of solving quadratic equations and other problems, but we also give you the practice you need to succeed over the long term. Need a full solution to a completing the square problem? Try our completing the square calculator. Ready to take your learning to the next level with “how” and “why” steps? Sign up for Cymath Plus today.