Find all values of $x$ such that $\dfrac{x}{x+4} = -\dfrac{9}{x+3}$. If you find more than one value, then list your solutions in increasing order, separated by commas.

Accepted Solution

We start with[tex]\dfrac{x}{x+4} = -\dfrac{9}{x+3}[/tex]Assuming that [tex]x \notin \{-4,\ -3\}[/tex] we can cross multiply the equation:[tex]x(x+3) = -9(x+4) \iff x^2+3x=-9x-36 \iff x^2+12x+36=0[/tex]You can recognize the perfect square pattern in the quadratic equation:[tex]x^2+12x+36 = (x)^2 + 2\cdot x \cdot 6 + (6)^2 = (x+6)^2[/tex]So, we have[tex]x^2+12x+36=0 \iff (x+6)^2 = 0 \iff x+6=0 \iff x=-6[/tex]Which is the only solution, with multiplicity 2.