Complete the Square

Learning Outcome

  • Complete the square to create a perfect square trinomial

We have seen the technique of completing the square before when solving quadratic equations. Using this method, we added or subtracted terms to both sides of the equation until we had a perfect square trinomial on one side of the equal sign.  Here is an example demonstrated earlier in the text.

We will use the example [latex]{x}^{2}+4x+1=0[/latex] to illustrate each step. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.

  1. Given a quadratic equation that cannot be factored and with [latex]a=1[/latex], first add or subtract the constant term to the right sign of the equal sign.
    [latex]{x}^{2}+4x=-1[/latex]
  2. Multiply the b term by [latex]\frac{1}{2}[/latex] and square it.
    [latex]\begin{array}{l}\frac{1}{2}\left(4\right)=2\hfill \\ {2}^{2}=4\hfill \end{array}[/latex]
  3. Add [latex]{\left(\frac{1}{2}b\right)}^{2}[/latex] to both sides of the equal sign and simplify the right side. We have
    [latex]\begin{array}{l}{x}^{2}+4x+4=-1+4\hfill \\ {x}^{2}+4x+4=3\hfill \end{array}[/latex]
  4. The left side of the equation can now be factored as a perfect square.
    [latex]\begin{array}{l}{x}^{2}+4x+4=3\hfill \\ {\left(x+2\right)}^{2}=3\hfill \end{array}[/latex]
  5. Use the square root property and solve.
    [latex]\begin{array}{l}\sqrt{{\left(x+2\right)}^{2}}=\pm \sqrt{3}\hfill \\ x+2=\pm \sqrt{3}\hfill \\ x=-2\pm \sqrt{3}\hfill \end{array}[/latex]
  6. The solutions are [latex]x=-2+\sqrt{3}[/latex], [latex]x=-2-\sqrt{3}[/latex].

We will see when studying conic sections that the method of completing the square comes in handy when rewriting the equation of a conic section given in general form. In preparation to use the method in this manner, it will be good to practice it first.

How To: use the method of complete the square to write a perfect square trinomial from an expression.

  1. Given an expression of the form [latex]a\left(x^2+bx\right)[/latex], add [latex]\left(\dfrac{b}{2}\right)^2[/latex] inside the parentheses.
  2. Then subtract [latex]a\left(\dfrac{b}{2}\right)^2[/latex] to counteract the change you made to the expression.
  3. If completing the square on one side of an equation, you may either subtract the value of [latex]a\left(\dfrac{b}{2}\right)^2[/latex] from that side, or add it to the other to maintain equality.
  4.  Then factor the perfect square trinomial you created inside the original parentheses.

The resulting form will look like this:

Given

[latex]\qquad a\left(x^2+bx\right)[/latex]

add [latex]\left(b/2\right)^2[/latex] inside the parentheses and subtract [latex]a\left(b/2\right)^2[/latex] to counteract the change you made to the expression

[latex]=a\left(x^2+bx+ \left(\dfrac{b}{2}\right)^2\right)-a\left(\dfrac{b}{2}\right)^2[/latex]

then factor the resulting perfect square trinomial

[latex]=a\left(x+ \dfrac{b}{2}\right)^2-a\left(\dfrac{b}{2}\right)^2[/latex].

Example : Create a perfect square trinomial using the method of complete the square

Complete the square on: [latex]3\left(x^2 - 10x\right)[/latex].

try it