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 $$\frac{8{p}^{2}}{2+p}=\frac{72}{5}$$?

Let's begin!

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

 1 Multiply both sides by $$2+p$$.$8{p}^{2}=\frac{72}{5}(2+p)$2 Simplify  $$\frac{72}{5}(2+p)$$  to  $$\frac{72(2+p)}{5}$$.$8{p}^{2}=\frac{72(2+p)}{5}$3 Multiply both sides by $$5$$.$40{p}^{2}=72(2+p)$4 Expand.$40{p}^{2}=144+72p$5 Move all terms to one side.$40{p}^{2}-144-72p=0$6 Factor out the common term $$8$$.$8(5{p}^{2}-18-9p)=0$7 Split the second term in $$5{p}^{2}-18-9p$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$5\times -18=-90$2 Ask: Which two numbers add up to $$-9$$ and multiply to $$-90$$?$$6$$ and $$-15$$3 Split $$-9p$$ as the sum of $$6p$$ and $$-15p$$.$5{p}^{2}+6p-15p-18$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$8(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))=0$9 Factor out the common term $$5p+6$$.$8(5p+6)(p-3)=0$10 Solve for $$p$$.1 Ask: When will $$(5p+6)(p-3)$$ equal zero?When $$5p+6=0$$ or $$p-3=0$$2 Solve each of the 2 equations above.$p=-\frac{6}{5},3$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$p=-\frac{6}{5},3$DoneDecimal Form: -1.2, 3p=-6/5,3