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

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

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

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, $$1+\sqrt{19},1-\sqrt{19},-10+i\sqrt{80},-10-i\sqrt{80}$$