Problem of the Week

Updated at May 20, 2024 12:15 PM

This week's problem comes from the equation category.

How can we solve the equation 8p22+p=725\frac{8{p}^{2}}{2+p}=\frac{72}{5}?

Let's begin!



8p22+p=725\frac{8{p}^{2}}{2+p}=\frac{72}{5}

1
Multiply both sides by 2+p2+p.
8p2=725(2+p)8{p}^{2}=\frac{72}{5}(2+p)

2
Simplify  725(2+p)\frac{72}{5}(2+p)  to  72(2+p)5\frac{72(2+p)}{5}.
8p2=72(2+p)58{p}^{2}=\frac{72(2+p)}{5}

3
Multiply both sides by 55.
40p2=72(2+p)40{p}^{2}=72(2+p)

4
Expand.
40p2=144+72p40{p}^{2}=144+72p

5
Move all terms to one side.
40p2−144−72p=040{p}^{2}-144-72p=0

6
Factor out the common term 88.
8(5p2−18−9p)=08(5{p}^{2}-18-9p)=0

7
Split the second term in 5p2−18−9p5{p}^{2}-18-9p into two terms.
8(5p2+6p−15p−18)=08(5{p}^{2}+6p-15p-18)=0

8
Factor out common terms in the first two terms, then in the last two terms.
8(p(5p+6)−3(5p+6))=08(p(5p+6)-3(5p+6))=0

9
Factor out the common term 5p+65p+6.
8(5p+6)(p−3)=08(5p+6)(p-3)=0

10
Solve for pp.
p=−65,3p=-\frac{6}{5},3

Done

Decimal Form: -1.2, 3
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