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.
\[16{x}^{2}-5x-64x+20=0\]

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 \(x\).
\[x=\frac{5}{16},4\]

Done

Decimal Form: 0.3125, 4
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