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.