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

Question

Isaiah Anthony · Accepted Answer

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.

Eva Adamson · Answer

undefined To solve the equation 5/(x+9) + 11/(x+2) = 9/(x^2 + 11x + 18), we can follow these steps:

1. Find a common denominator for all the fractions involved. In this case, the common denominator is (x+9)(x+2).

2. Multiply each term by the common denominator to eliminate the fractions.

3. Simplify the equation by distributing and combining like terms.

4. Rearrange the equation to bring all terms to one side and set it equal to zero.

5. Factor the quadratic expression (x^2 + 11x + 18) and set it equal to zero.

6. Solve the resulting equation(s) to find the possible values of x.

By following these steps, you can find the solution(s) to the given equation.