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Section 2-6 : Quadratic Equations - Part II

2. Complete the square on the following expression.

${u^2} - 11u$

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First, we need to identify the number we need to add to this. Recall that we will need the coefficient of the $$u$$ to do this. The number we need is,

${\left( {\frac{{ - 11}}{2}} \right)^2} = \frac{{{{\left( { - 11} \right)}^2}}}{{{{\left( 2 \right)}^2}}} = \frac{{121}}{4}$ Show Step 2

To complete the square all we need to do then is add this to the expression and factor the result. Doing this gives,

$\require{color}\require{bbox} \bbox[2pt,border:1px solid black]{{{u^2} - 11u \,{\color{Red} + \frac{{121}}{4}} = {{\left( {u - \frac{{11}}{2}} \right)}^2}}}$

Recall that this will always factor as $$u$$ plus the number inside the parenthesis in the first step, $$- \frac{{11}}{2}$$ in this case.

Do not get too excited about the fractions that can show up in these problems. They will be there occasionally and so we need to be able to deal with them. Luckily, if you can recall the “trick” to the factoring they aren’t all that bad.