Problem of the Week

Updated at May 2, 2022 3:27 PM

How can we solve the equation 4(3−y)2(y+2)=244{(3-y)}^{2}(y+2)=24?

Below is the solution.



4(3−y)2(y+2)=244{(3-y)}^{2}(y+2)=24

1
Expand.
36y+72−24y2−48y+4y3+8y2=2436y+72-24{y}^{2}-48y+4{y}^{3}+8{y}^{2}=24

2
Simplify  36y+72−24y2−48y+4y3+8y236y+72-24{y}^{2}-48y+4{y}^{3}+8{y}^{2}  to  −12y+72−16y2+4y3-12y+72-16{y}^{2}+4{y}^{3}.
−12y+72−16y2+4y3=24-12y+72-16{y}^{2}+4{y}^{3}=24

3
Move all terms to one side.
12y−72+16y2−4y3+24=012y-72+16{y}^{2}-4{y}^{3}+24=0

4
Simplify  12y−72+16y2−4y3+2412y-72+16{y}^{2}-4{y}^{3}+24  to  12y−48+16y2−4y312y-48+16{y}^{2}-4{y}^{3}.
12y−48+16y2−4y3=012y-48+16{y}^{2}-4{y}^{3}=0

5
Factor out the common term 44.
4(3y−12+4y2−y3)=04(3y-12+4{y}^{2}-{y}^{3})=0

6
Factor 3y−12+4y2−y33y-12+4{y}^{2}-{y}^{3} using Polynomial Division.
4(−y2+3)(y−4)=04(-{y}^{2}+3)(y-4)=0

7
Solve for yy.
y=4,±3y=4,\pm \sqrt{3}

Done

Decimal Form: 4, ±1.732051
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