Solve x^2+mx+n/x^2-7x+10=x+1/x-5Rightarrowm+n=?

$$\frac { x ^ { 2 } + m x + n } { x ^ { 2 } - 7 x + 10 } = \frac { x + 1 } { x - 5 } \Rightarrow m + n = ?$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{x+n+2}{x}\text{, }&x\neq 0\text{ and }x\neq 2\text{ and }x\neq 5\\m\in \mathrm{C}\text{, }&x=0\text{ and }n=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x^{2}-7x+10,x-5.

$$x^{2}+mx+n=\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$x^{2}+mx+n=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$mx+n=x^{2}-x-2-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$mx+n=-x-2$$

Subtract n from both sides.

$$mx=-x-2-n$$

The equation is in standard form.

$$xm=-x-n-2$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-x-n-2}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-x-n-2}{x}$$

Divide -x-2-n by x.

$$m=-\frac{x+n+2}{x}$$

Solve for n (complex solution)

$$n=-\left(mx+x+2\right)$$

$$x\neq 2\text{ and }x\neq 5$$

Solution Steps

Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x^{2}-7x+10,x-5.

$$x^{2}+mx+n=\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$x^{2}+mx+n=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$mx+n=x^{2}-x-2-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$mx+n=-x-2$$

Subtract mx from both sides.

$$n=-x-2-mx$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{x+n+2}{x}\text{, }&x\neq 0\text{ and }x\neq 5\text{ and }x\neq 2\\m\in \mathrm{R}\text{, }&x=0\text{ and }n=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x^{2}-7x+10,x-5.

$$x^{2}+mx+n=\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$x^{2}+mx+n=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$mx+n=x^{2}-x-2-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$mx+n=-x-2$$

Subtract n from both sides.

$$mx=-x-2-n$$

The equation is in standard form.

$$xm=-x-n-2$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-x-n-2}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-x-n-2}{x}$$

Divide -x-2-n by x.

$$m=-\frac{x+n+2}{x}$$

Solve for n

$$n=-\left(mx+x+2\right)$$

$$x\neq 5\text{ and }x\neq 2$$

Solution Steps

Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x^{2}-7x+10,x-5.

$$x^{2}+mx+n=\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$x^{2}+mx+n=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$mx+n=x^{2}-x-2-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$mx+n=-x-2$$

Subtract mx from both sides.

$$n=-x-2-mx$$