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.
+26367 Find all values of x such that
\(\dfrac{x}{x+4} = -\dfrac{9}{x+3}\)
\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.
\(\text{ $x\ne -4$ and $x\ne -3$ }\)
\(\begin{array}{|rcll|} \hline \dfrac{x}{x+4} &=& -\dfrac{9}{x+3} \quad & | \quad \cdot (x+3) \\\\ \dfrac{x(x+3)}{x+4} &=& -9 \quad & | \quad \cdot (x+4) \\\\ x(x+3) &=& -9(x+4) \\\\ x^2+3x &=& -9x -36 \quad & | \quad +9x \\\\ x^2+12x &=& -36 \quad & | \quad +36 \\\\ x^2+12x+36 &=&0 \\\\ x &=& \dfrac{-12\pm \sqrt{12^2 -4\cdot 36 } }{2} \\\\ x &=& \dfrac{-12\pm \sqrt{144 -144 } }{2} \\\\ x &=& \dfrac{-12\pm \sqrt{0} }{2} \\\\ x &=& \dfrac{-12\pm 0 }{2} \\\\ x &=& \dfrac{-12}{2} \\\\ \mathbf{x} &\mathbf{=}& \mathbf{-6} \\ \hline \end{array}\)
