# What is the solution to the equation below? [duplicate]

This question already has an answer here:

Solve the equation below.

$$x^2+\frac{81x^2}{(9+x)^2}=40$$

I couldn't solve it after trying many time.

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## marked as duplicate by YACP, Stefan Hansen, vonbrand, Paul, Davide GiraudoMar 27 '13 at 10:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

differential equations? –  user52188 Jan 15 '13 at 4:25
First you clear out the denominator, giving you a quartic polynomial equation. In principle it can be factored analytically... in practice, unless there's an obvious factorization, you use a numerical method to find the roots. –  user7530 Jan 15 '13 at 4:34

## 2 Answers

Here is the answer computed in Wolfram|Alpha. Note that there is a feature on this site, called "Step-by-step solution," that allows you to see precisely how they arrived at the answer.

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Please try to describe as much here as possible in order to make the answer self-contained. Links are fine as support, but they can go stale and then an answer which is nothing more than a link loses its value. –  robjohn May 23 '13 at 6:17

Multiplying both sides by $(x+9)^2$ yields $$(x^2+18x+162)x^2=40(x^2+18x+81)$$ which is equivalent to $$x^4+18x^3+122x^2-720x-3240=0$$ Factorization gives $$x^4+18x^3+122x^2-720x-3240=(x^2+20x+180)(x^2-2x-18)$$ Which has roots, given by the quadratic formula, $$1+\sqrt{19},1-\sqrt{19},-10+i\sqrt{80},-10-i\sqrt{80}$$

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