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Solve for m (complex solution)
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Solve for n (complex solution)
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Solve for m
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Solve for n
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x^{2}+mx+n=\left(x-2\right)\left(x+1\right)
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=x^{2}-x-2
Use the distributive property to multiply x-2 by x+1 and combine like terms.
mx+n=x^{2}-x-2-x^{2}
Subtract x^{2} from both sides.
mx+n=-x-2
Combine x^{2} and -x^{2} to get 0.
mx=-x-2-n
Subtract n from both sides.
xm=-x-n-2
The equation is in standard form.
\frac{xm}{x}=\frac{-x-n-2}{x}
Divide both sides by x.
m=\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.
x^{2}+mx+n=\left(x-2\right)\left(x+1\right)
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=x^{2}-x-2
Use the distributive property to multiply x-2 by x+1 and combine like terms.
mx+n=x^{2}-x-2-x^{2}
Subtract x^{2} from both sides.
mx+n=-x-2
Combine x^{2} and -x^{2} to get 0.
n=-x-2-mx
Subtract mx from both sides.
x^{2}+mx+n=\left(x-2\right)\left(x+1\right)
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=x^{2}-x-2
Use the distributive property to multiply x-2 by x+1 and combine like terms.
mx+n=x^{2}-x-2-x^{2}
Subtract x^{2} from both sides.
mx+n=-x-2
Combine x^{2} and -x^{2} to get 0.
mx=-x-2-n
Subtract n from both sides.
xm=-x-n-2
The equation is in standard form.
\frac{xm}{x}=\frac{-x-n-2}{x}
Divide both sides by x.
m=\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.
x^{2}+mx+n=\left(x-2\right)\left(x+1\right)
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=x^{2}-x-2
Use the distributive property to multiply x-2 by x+1 and combine like terms.
mx+n=x^{2}-x-2-x^{2}
Subtract x^{2} from both sides.
mx+n=-x-2
Combine x^{2} and -x^{2} to get 0.
n=-x-2-mx
Subtract mx from both sides.