Solve left(2+mright)m/left(1+mright)^2=0*22m+1*28/m+1

$$\frac{ \left( 2+m \right) m }{ { \left(1+m \right) }^{ 2 } } = \frac{ 0 \cdot 22m+1 \cdot 28 }{ m+1 }$$

Solve for m

$$m=\sqrt{197}+13\approx 27.035668848$$

$$m=13-\sqrt{197}\approx -1.035668848$$

Steps Using the Quadratic Formula

Variable m cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(m+1\right)^{2}, the least common multiple of \left(1+m\right)^{2},m+1.

$$\left(2+m\right)m=\left(m+1\right)\left(0\times 22m+1\times 28\right)$$

Use the distributive property to multiply 2+m by m.

$$2m+m^{2}=\left(m+1\right)\left(0\times 22m+1\times 28\right)$$

Do the multiplications.

$$2m+m^{2}=\left(m+1\right)\left(0m+28\right)$$

Anything times zero gives zero.

$$2m+m^{2}=\left(m+1\right)\left(0+28\right)$$

Add 0 and 28 to get 28.

$$2m+m^{2}=\left(m+1\right)\times 28$$

Use the distributive property to multiply m+1 by 28.

$$2m+m^{2}=28m+28$$

Subtract 28m from both sides.

$$2m+m^{2}-28m=28$$

Combine 2m and -28m to get -26m.

$$-26m+m^{2}=28$$

Subtract 28 from both sides.

$$-26m+m^{2}-28=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$m^{2}-26m-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -26 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-28\right)}}{2}$$

Square -26.

$$m=\frac{-\left(-26\right)±\sqrt{676-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$m=\frac{-\left(-26\right)±\sqrt{676+112}}{2}$$

Add 676 to 112.

$$m=\frac{-\left(-26\right)±\sqrt{788}}{2}$$

Take the square root of 788.

$$m=\frac{-\left(-26\right)±2\sqrt{197}}{2}$$

The opposite of -26 is 26.

$$m=\frac{26±2\sqrt{197}}{2}$$

Now solve the equation m=\frac{26±2\sqrt{197}}{2} when ± is plus. Add 26 to 2\sqrt{197}.

$$m=\frac{2\sqrt{197}+26}{2}$$

Divide 26+2\sqrt{197} by 2.

$$m=\sqrt{197}+13$$

Now solve the equation m=\frac{26±2\sqrt{197}}{2} when ± is minus. Subtract 2\sqrt{197} from 26.

$$m=\frac{26-2\sqrt{197}}{2}$$

Divide 26-2\sqrt{197} by 2.

$$m=13-\sqrt{197}$$

The equation is now solved.

$$m=\sqrt{197}+13 m=13-\sqrt{197}$$

Steps for Completing the Square

Variable m cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(m+1\right)^{2}, the least common multiple of \left(1+m\right)^{2},m+1.

$$\left(2+m\right)m=\left(m+1\right)\left(0\times 22m+1\times 28\right)$$

Use the distributive property to multiply 2+m by m.

$$2m+m^{2}=\left(m+1\right)\left(0\times 22m+1\times 28\right)$$

Do the multiplications.

$$2m+m^{2}=\left(m+1\right)\left(0m+28\right)$$

Anything times zero gives zero.

$$2m+m^{2}=\left(m+1\right)\left(0+28\right)$$

Add 0 and 28 to get 28.

$$2m+m^{2}=\left(m+1\right)\times 28$$

Use the distributive property to multiply m+1 by 28.

$$2m+m^{2}=28m+28$$

Subtract 28m from both sides.

$$2m+m^{2}-28m=28$$

Combine 2m and -28m to get -26m.

$$-26m+m^{2}=28$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$m^{2}-26m=28$$

Divide -26, the coefficient of the x term, by 2 to get -13. Then add the square of -13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-26m+\left(-13\right)^{2}=28+\left(-13\right)^{2}$$

Square -13.

$$m^{2}-26m+169=28+169$$

Add 28 to 169.

$$m^{2}-26m+169=197$$

Factor m^{2}-26m+169. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-13\right)^{2}=197$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-13\right)^{2}}=\sqrt{197}$$

Simplify.

$$m-13=\sqrt{197} m-13=-\sqrt{197}$$

Add 13 to both sides of the equation.

$$m=\sqrt{197}+13 m=13-\sqrt{197}$$