Solve m(x)=x^2-6x+8/x^2+5x-14,w

$$m ( x ) = \frac { x ^ { 2 } - 6 x + 8 } { x ^ { 2 } + 5 x - 14 } , w$$

Solve for m

$$m=-\frac{4-x}{x\left(x+7\right)}$$

$$x\neq -7\text{ and }x\neq 0\text{ and }x\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+7\right).

$$mx\left(x-2\right)\left(x+7\right)=x^{2}-6x+8$$

Use the distributive property to multiply mx by x-2.

$$\left(mx^{2}-2mx\right)\left(x+7\right)=x^{2}-6x+8$$

Use the distributive property to multiply mx^{2}-2mx by x+7 and combine like terms.

$$mx^{3}+5mx^{2}-14mx=x^{2}-6x+8$$

Combine all terms containing m.

$$\left(x^{3}+5x^{2}-14x\right)m=x^{2}-6x+8$$

Divide both sides by x^{3}+5x^{2}-14x.

$$\frac{\left(x^{3}+5x^{2}-14x\right)m}{x^{3}+5x^{2}-14x}=\frac{\left(x-4\right)\left(x-2\right)}{x^{3}+5x^{2}-14x}$$

Dividing by x^{3}+5x^{2}-14x undoes the multiplication by x^{3}+5x^{2}-14x.

$$m=\frac{\left(x-4\right)\left(x-2\right)}{x^{3}+5x^{2}-14x}$$

Divide \left(-4+x\right)\left(-2+x\right) by x^{3}+5x^{2}-14x.

$$m=\frac{x-4}{x\left(x+7\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{49m^{2}-30m+1}-7m+1}{2m}\text{, }&m\neq 0\text{ and }m\neq -\frac{1}{9}\\x=\frac{\sqrt{49m^{2}-30m+1}-7m+1}{2m}\text{, }&m\neq 0\\x=4\text{, }&m=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{49m^{2}-30m+1}-7m+1}{2m}\text{, }&\left(m\neq -\frac{1}{9}\text{ and }m\neq 0\text{ and }m\leq \frac{15-4\sqrt{11}}{49}\right)\text{ or }m\geq \frac{4\sqrt{11}+15}{49}\\x=\frac{\sqrt{49m^{2}-30m+1}-7m+1}{2m}\text{, }&\left(m\neq 0\text{ and }m\leq \frac{15-4\sqrt{11}}{49}\right)\text{ or }m\geq \frac{4\sqrt{11}+15}{49}\\x=4\text{, }&m=0\end{matrix}\right.$$