Problem of the Week

Updated at Sep 21, 2020 10:00 AM

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Mar 24, 2025

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}

Multiply both sides by

4x

16x+16{x}^{2}+20=85x

Move all terms to one side.

16x+16{x}^{2}+20-85x=0

Simplify

16x+16{x}^{2}+20-85x

-69x+16{x}^{2}+20

-69x+16{x}^{2}+20=0

Split the second term in

-69x+16{x}^{2}+20

into two terms.

Multiply the coefficient of the first term by the constant term.

16\times 20=320

Ask: Which two numbers add up to

-69

and multiply to

320

-5

and

-64

Split

-69x

as the sum of

-5x

and

-64x

16{x}^{2}-5x-64x+20

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16{x}^{2}-5x-64x+20=0

Factor out common terms in the first two terms, then in the last two terms.

x(16x-5)-4(16x-5)=0

Factor out the common term

16x-5

(16x-5)(x-4)=0

Solve for

x

Ask: When will

(16x-5)(x-4)

equal zero?

When

16x-5=0

x-4=0

Solve each of the 2 equations above.

x=\frac{5}{16},4

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x=\frac{5}{16},4

Done

Decimal Form: 0.3125, 4