PCC SLC Math Resources

Completing the Square.

We frequently have to do some preliminary work to have an equation of form \(u^2=k\text{.}\) The process involves completing the square. Completing the square entails adding a number to an expression of form \(x^2+bx\) so that the resultant trinomial factors into a perfect square. The number that complete the square is always the square of half of \(b\text{.}\)

For example, the number that completes the square for \(x^2+10x\) is \(5^2\) which is \(25\text{.}\) Note:

The steps followed when using the square root method are listed below.

1. Perform any and all manipulations so that the equation has the form \(x^2+bx=c\text{.}\)
2. Complete the square by adding \(\left(\frac{b}{2}\right)^2\text{.}\) Note that to keep the equation in balance \(\left(\frac{b}{2}\right)^2\) also needs to be added to the other side of the equation.
3. Factor the non-constant side of the equation. The equation should now have form \(u^2=k\text{.}\)
4. If the constant, \(k\text{,}\) is negative, then the equation has no solution. If \(k\) is zero, then the equation has one solution which is determined by solving \(u=0\text{.}\) Step 5 is written under the assumption that \(k\) is a positive number.
5. Invoke the square root property, i.e. \(u=\pm \sqrt{k}\text{.}\) If \(k\) is a perfect square, you need to write out and solve two equations.
   \begin{equation*} u=-\sqrt{k}\text{ or }u=\sqrt{k}\text{.} \end{equation*}
   If \(k\) is not a perfect square, you may solve for \(u\) maintaining the plus/minus sign. Make sure to simplify the square root should it simplify.
6. State your solutions and/or solution set.

Several examples follow.

You can use Figure 12.4.5 to generate several more examples/practice problems. You'll need to write the steps down, because only one step is shown at a time.