Section 2.6 : Quadratic Equations - Part II
3. Complete the square on the following expression.
\[2{z^2} - 12z\]Show All Steps Hide All Steps
Start SolutionRemember that prior to completing the square we need a coefficient of one on the squared variable. However, we can’t just “cancel” it since that requires an equation which we don’t have.
Therefore, we need to first factor a 2 out of the expression as follows,
\[2{z^2} - 12z = 2\left( {{z^2} - 6z} \right)\]We can now proceed with completing the square on the expression inside the parenthesis.
Show Step 2Next, we’ll need the number we need to add onto the expression inside the parenthesis. We’ll need the coefficient of the \(z\) to do this. The number we need is,
\[{\left( {\frac{{ - 6}}{2}} \right)^2} = {\left( { - 3} \right)^2} = 9\] Show Step 3To complete the square all we need to do then is add this to the expression inside the parenthesis and factor the result. Doing this gives,
\[\require{color}\require{bbox} \bbox[2pt,border:1px solid black]{{2{z^2} - 12z = 2\left( {{z^2} - 6z \,{\color{Red} + 9}} \right) = 2{{\left( {z - 3} \right)}^2}}}\]Be careful when the coefficient of the squared term is not a one! In order to get the correct answer to completing the square we must have a coefficient of one on the squared term!