How do you solve $5/(x+9) + 11 /( x+2) =9 /( x ^2 + 11x + 18)$ ?

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Answer 1 · 2015-10-09T21:47:25

Solution: $x=-25/4$ .

First of all, compute the GCD in the first member:

$5/(x+9) + 11/(x+2) = (5(x+2) + 11(x+9))/((x+9)(x+2))$

Simplifying the numerator, we obtain

$5x+10+11x+99=16x+109$

Simplifying the denominator, we obtain

$x^2+2x+9x+18 = x^2 +11x+18$

So, our equation becomes

$(16x+109)/(x^2 +11x+18) = 9/(x^2 +11x+18)$

Since the denominators are equal, the equality holds if and only if it holds between the numerators, i.e.

$16x+109 = 9 \iff 16x = -100 \iff x=-100/16 = -25/4$

P.S.: the denominator(s) are zero for $x=-2$ or $x=-9$ , so the root we found is acceptable.

How do you solve 5/(x+9) + 11 /( x+2) =9 /( x ^2 + 11x + 18) 5/(x+9) + 11 /( x+2) =9 /( x ^2 + 11x + 18)?