# Problem of the Week

## Updated at Oct 3, 2022 1:58 PM

This week we have another equation problem:

How would you solve the equation $$\frac{4(2+t)}{2+{t}^{2}}=\frac{20}{11}$$?

Let's start!

$\frac{4(2+t)}{2+{t}^{2}}=\frac{20}{11}$

 1 Multiply both sides by $$2+{t}^{2}$$.$4(2+t)=\frac{20}{11}(2+{t}^{2})$2 Simplify  $$\frac{20}{11}(2+{t}^{2})$$  to  $$\frac{20(2+{t}^{2})}{11}$$.$4(2+t)=\frac{20(2+{t}^{2})}{11}$3 Multiply both sides by $$11$$.$44(2+t)=20(2+{t}^{2})$4 Expand.$88+44t=40+20{t}^{2}$5 Move all terms to one side.$88+44t-40-20{t}^{2}=0$6 Simplify  $$88+44t-40-20{t}^{2}$$  to  $$48+44t-20{t}^{2}$$.$48+44t-20{t}^{2}=0$7 Factor out the common term $$4$$.$4(12+11t-5{t}^{2})=0$8 Factor out the negative sign.$4\times -(5{t}^{2}-11t-12)=0$9 Divide both sides by $$4$$.$-5{t}^{2}+11t+12=0$10 Multiply both sides by $$-1$$.$5{t}^{2}-11t-12=0$11 Split the second term in $$5{t}^{2}-11t-12$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$5\times -12=-60$2 Ask: Which two numbers add up to $$-11$$ and multiply to $$-60$$?$$4$$ and $$-15$$3 Split $$-11t$$ as the sum of $$4t$$ and $$-15t$$.$5{t}^{2}+4t-15t-12$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$5{t}^{2}+4t-15t-12=0$12 Factor out common terms in the first two terms, then in the last two terms.$t(5t+4)-3(5t+4)=0$13 Factor out the common term $$5t+4$$.$(5t+4)(t-3)=0$14 Solve for $$t$$.1 Ask: When will $$(5t+4)(t-3)$$ equal zero?When $$5t+4=0$$ or $$t-3=0$$2 Solve each of the 2 equations above.$t=-\frac{4}{5},3$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$t=-\frac{4}{5},3$DoneDecimal Form: -0.8, 3t=-4/5,3