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### Section 2.6 : Quadratic Equations - Part II

3. Complete the square on the following expression.

$2{z^2} - 12z$

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Start Solution

Remember 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 2

Next, 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 3

To 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!