Solve m^2+11m+30/4m^2-28m-120=m+10/4m-40+m/4m+12

$$\frac { m ^ { 2 } + 11 m + 30 } { 4 m ^ { 2 } - 28 m - 120 } = \frac { m + 10 } { 4 m - 40 } + \frac { m } { 4 m + 12 }$$

Solve for m

$$m=8$$

$$m=0$$

Steps Using Factoring

Variable m cannot be equal to any of the values -3,10 since division by zero is not defined. Multiply both sides of the equation by 4\left(m-10\right)\left(m+3\right), the least common multiple of 4m^{2}-28m-120,4m-40,4m+12.

$$m^{2}+11m+30=\left(m+3\right)\left(m+10\right)+\left(m-10\right)m$$

Use the distributive property to multiply m+3 by m+10 and combine like terms.

$$m^{2}+11m+30=m^{2}+13m+30+\left(m-10\right)m$$

Use the distributive property to multiply m-10 by m.

$$m^{2}+11m+30=m^{2}+13m+30+m^{2}-10m$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$m^{2}+11m+30=2m^{2}+13m+30-10m$$

Combine 13m and -10m to get 3m.

$$m^{2}+11m+30=2m^{2}+3m+30$$

Subtract 2m^{2} from both sides.

$$m^{2}+11m+30-2m^{2}=3m+30$$

Combine m^{2} and -2m^{2} to get -m^{2}.

$$-m^{2}+11m+30=3m+30$$

Subtract 3m from both sides.

$$-m^{2}+11m+30-3m=30$$

Combine 11m and -3m to get 8m.

$$-m^{2}+8m+30=30$$

Subtract 30 from both sides.

$$-m^{2}+8m+30-30=0$$

Subtract 30 from 30 to get 0.

$$-m^{2}+8m=0$$

Factor out m.

$$m\left(-m+8\right)=0$$

To find equation solutions, solve m=0 and -m+8=0.

$$m=0 m=8$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values -3,10 since division by zero is not defined. Multiply both sides of the equation by 4\left(m-10\right)\left(m+3\right), the least common multiple of 4m^{2}-28m-120,4m-40,4m+12.

$$m^{2}+11m+30=\left(m+3\right)\left(m+10\right)+\left(m-10\right)m$$

Use the distributive property to multiply m+3 by m+10 and combine like terms.

$$m^{2}+11m+30=m^{2}+13m+30+\left(m-10\right)m$$

Use the distributive property to multiply m-10 by m.

$$m^{2}+11m+30=m^{2}+13m+30+m^{2}-10m$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$m^{2}+11m+30=2m^{2}+13m+30-10m$$

Combine 13m and -10m to get 3m.

$$m^{2}+11m+30=2m^{2}+3m+30$$

Subtract 2m^{2} from both sides.

$$m^{2}+11m+30-2m^{2}=3m+30$$

Combine m^{2} and -2m^{2} to get -m^{2}.

$$-m^{2}+11m+30=3m+30$$

Subtract 3m from both sides.

$$-m^{2}+11m+30-3m=30$$

Combine 11m and -3m to get 8m.

$$-m^{2}+8m+30=30$$

Subtract 30 from both sides.

$$-m^{2}+8m+30-30=0$$

Subtract 30 from 30 to get 0.

$$-m^{2}+8m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-8±\sqrt{8^{2}}}{2\left(-1\right)}$$

Take the square root of 8^{2}.

$$m=\frac{-8±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-8±8}{-2}$$

Now solve the equation m=\frac{-8±8}{-2} when ± is plus. Add -8 to 8.

$$m=\frac{0}{-2}$$

Divide 0 by -2.

$$m=0$$

Now solve the equation m=\frac{-8±8}{-2} when ± is minus. Subtract 8 from -8.

$$m=-\frac{16}{-2}$$

Divide -16 by -2.

$$m=8$$

The equation is now solved.

$$m=0 m=8$$

Steps for Completing the Square

Variable m cannot be equal to any of the values -3,10 since division by zero is not defined. Multiply both sides of the equation by 4\left(m-10\right)\left(m+3\right), the least common multiple of 4m^{2}-28m-120,4m-40,4m+12.

$$m^{2}+11m+30=\left(m+3\right)\left(m+10\right)+\left(m-10\right)m$$

Use the distributive property to multiply m+3 by m+10 and combine like terms.

$$m^{2}+11m+30=m^{2}+13m+30+\left(m-10\right)m$$

Use the distributive property to multiply m-10 by m.

$$m^{2}+11m+30=m^{2}+13m+30+m^{2}-10m$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$m^{2}+11m+30=2m^{2}+13m+30-10m$$

Combine 13m and -10m to get 3m.

$$m^{2}+11m+30=2m^{2}+3m+30$$

Subtract 2m^{2} from both sides.

$$m^{2}+11m+30-2m^{2}=3m+30$$

Combine m^{2} and -2m^{2} to get -m^{2}.

$$-m^{2}+11m+30=3m+30$$

Subtract 3m from both sides.

$$-m^{2}+11m+30-3m=30$$

Combine 11m and -3m to get 8m.

$$-m^{2}+8m+30=30$$

Subtract 30 from both sides.

$$-m^{2}+8m=30-30$$

Subtract 30 from 30 to get 0.

$$-m^{2}+8m=0$$

Divide both sides by -1.

$$\frac{-m^{2}+8m}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\frac{8}{-1}m=\frac{0}{-1}$$

Divide 8 by -1.

$$m^{2}-8m=\frac{0}{-1}$$

Divide 0 by -1.

$$m^{2}-8m=0$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-8m+\left(-4\right)^{2}=\left(-4\right)^{2}$$

Square -4.

$$m^{2}-8m+16=16$$

Factor m^{2}-8m+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-4\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-4\right)^{2}}=\sqrt{16}$$

Simplify.

$$m-4=4 m-4=-4$$

Add 4 to both sides of the equation.

$$m=8 m=0$$