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Section 4-2 : Parabolas

8. Convert the following equations into the form \(y = a{\left( {x - h} \right)^2} + k\).

\[f\left( x \right) = {x^2} - 24x + 157\]

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Start Solution

We’ll need to do the modified completing the square process described in the notes for this section.

The first step in this process is to make sure that we have a coefficient of one on the \({x^2}\), which we already have, so there is nothing we need to do in that regards for this problem.

Show Step 2

Next, we need to take one-half the coefficient of the \(x\), square it and then add and subtract it onto the equation.

\[{\left( {\frac{{ - 24}}{2}} \right)^2} = {\left( { - 12} \right)^2} = 144\] \[\require{color}f\left( x \right) = {x^2} - 24x \,{\color{Red} + 144 - 144} + 157\] Show Step 3

Finally, all we need to do is factor the first three terms and combine the last two numbers to get,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( x \right) = {{\left( {x - 12} \right)}^2} + 13}}\]