Problem of the Week

Updated at Jul 31, 2023 10:17 AM

How would you solve \({(3-y)}^{2}(4+4y)=32\)?

Below is the solution.



\[{(3-y)}^{2}(4+4y)=32\]

1
Expand.
\[36+36y-24y-24{y}^{2}+4{y}^{2}+4{y}^{3}=32\]

2
Simplify  \(36+36y-24y-24{y}^{2}+4{y}^{2}+4{y}^{3}\)  to  \(36+12y-20{y}^{2}+4{y}^{3}\).
\[36+12y-20{y}^{2}+4{y}^{3}=32\]

3
Move all terms to one side.
\[36+12y-20{y}^{2}+4{y}^{3}-32=0\]

4
Simplify  \(36+12y-20{y}^{2}+4{y}^{3}-32\)  to  \(4+12y-20{y}^{2}+4{y}^{3}\).
\[4+12y-20{y}^{2}+4{y}^{3}=0\]

5
Factor out the common term \(4\).
\[4(1+3y-5{y}^{2}+{y}^{3})=0\]

6
Factor \(1+3y-5{y}^{2}+{y}^{3}\) using Polynomial Division.
\[4({y}^{2}-4y-1)(y-1)=0\]

7
Solve for \(y\).
\[y=1\]

8
Use the Quadratic Formula.
\[y=\frac{4+2\sqrt{5}}{2},\frac{4-2\sqrt{5}}{2}\]

9
Collect all solutions from the previous steps.
\[y=1,\frac{4+2\sqrt{5}}{2},\frac{4-2\sqrt{5}}{2}\]

10
Simplify solutions.
\[y=1,2+\sqrt{5},2-\sqrt{5}\]

Done

Decimal Form: 1, 4.236068, -0.236068
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