Problem of the Week

Updated at Jul 15, 2024 2:53 PM

This week we have another equation problem:

How would you solve $$\frac{4q+2}{{(q-3)}^{2}}=10$$?

Let's start!

$\frac{4q+2}{{(q-3)}^{2}}=10$

 1 Factor out the common term $$2$$.$\frac{2(2q+1)}{{(q-3)}^{2}}=10$2 Multiply both sides by $${(q-3)}^{2}$$.$2(2q+1)=10{(q-3)}^{2}$3 Divide both sides by $$2$$.$2q+1=5{(q-3)}^{2}$4 Expand.$2q+1=5{q}^{2}-30q+45$5 Move all terms to one side.$2q+1-5{q}^{2}+30q-45=0$6 Simplify  $$2q+1-5{q}^{2}+30q-45$$  to  $$32q-44-5{q}^{2}$$.$32q-44-5{q}^{2}=0$7 Multiply both sides by $$-1$$.$5{q}^{2}-32q+44=0$8 Split the second term in $$5{q}^{2}-32q+44$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$5\times 44=220$2 Ask: Which two numbers add up to $$-32$$ and multiply to $$220$$?$$-10$$ and $$-22$$3 Split $$-32q$$ as the sum of $$-10q$$ and $$-22q$$.$5{q}^{2}-10q-22q+44$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$5{q}^{2}-10q-22q+44=0$9 Factor out common terms in the first two terms, then in the last two terms.$5q(q-2)-22(q-2)=0$10 Factor out the common term $$q-2$$.$(q-2)(5q-22)=0$11 Solve for $$q$$.1 Ask: When will $$(q-2)(5q-22)$$ equal zero?When $$q-2=0$$ or $$5q-22=0$$2 Solve each of the 2 equations above.$q=2,\frac{22}{5}$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$q=2,\frac{22}{5}$DoneDecimal Form: 2, 4.4q=2,22/5