# Problem of the Week

## Updated at Sep 21, 2020 10:00 AM

This week's problem comes from the equation category.

How can we solve the equation $$4+4x+\frac{5}{x}=\frac{85}{4}$$?

Let's begin!

$4+4x+\frac{5}{x}=\frac{85}{4}$

 1 Multiply both sides by $$4x$$.$16x+16{x}^{2}+20=85x$2 Move all terms to one side.$16x+16{x}^{2}+20-85x=0$3 Simplify  $$16x+16{x}^{2}+20-85x$$  to  $$-69x+16{x}^{2}+20$$.$-69x+16{x}^{2}+20=0$4 Split the second term in $$-69x+16{x}^{2}+20$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$16\times 20=320$2 Ask: Which two numbers add up to $$-69$$ and multiply to $$320$$?$$-5$$ and $$-64$$3 Split $$-69x$$ as the sum of $$-5x$$ and $$-64x$$.$16{x}^{2}-5x-64x+20$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$16{x}^{2}-5x-64x+20=0$5 Factor out common terms in the first two terms, then in the last two terms.$x(16x-5)-4(16x-5)=0$6 Factor out the common term $$16x-5$$.$(16x-5)(x-4)=0$7 Solve for $$x$$.1 Ask: When will $$(16x-5)(x-4)$$ equal zero?When $$16x-5=0$$ or $$x-4=0$$2 Solve each of the 2 equations above.$x=\frac{5}{16},4$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$x=\frac{5}{16},4$Done Decimal Form: 0.3125, 4x=5/16,4