Solve m(m+2)/m-1=1/2

$$\frac { m ( m + 2 ) } { m - 1 } = \frac { 1 } { 2 }$$

Solve for m

$$m=-\frac{1}{2}=-0.5$$

$$m=-1$$

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 2\left(m-1\right), the least common multiple of m-1,2.

$$2m\left(m+2\right)=m-1$$

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

$$2m^{2}+4m=m-1$$

Subtract m from both sides.

$$2m^{2}+4m-m=-1$$

Combine 4m and -m to get 3m.

$$2m^{2}+3m=-1$$

Add 1 to both sides.

$$2m^{2}+3m+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-3±\sqrt{3^{2}-4\times 2}}{2\times 2}$$

Square 3.

$$m=\frac{-3±\sqrt{9-4\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-3±\sqrt{9-8}}{2\times 2}$$

Add 9 to -8.

$$m=\frac{-3±\sqrt{1}}{2\times 2}$$

Take the square root of 1.

$$m=\frac{-3±1}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-3±1}{4}$$

Now solve the equation m=\frac{-3±1}{4} when ± is plus. Add -3 to 1.

$$m=-\frac{2}{4}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$m=-\frac{1}{2}$$

Now solve the equation m=\frac{-3±1}{4} when ± is minus. Subtract 1 from -3.

$$m=-\frac{4}{4}$$

Divide -4 by 4.

$$m=-1$$

The equation is now solved.

$$m=-\frac{1}{2} m=-1$$

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 2\left(m-1\right), the least common multiple of m-1,2.

$$2m\left(m+2\right)=m-1$$

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

$$2m^{2}+4m=m-1$$

Subtract m from both sides.

$$2m^{2}+4m-m=-1$$

Combine 4m and -m to get 3m.

$$2m^{2}+3m=-1$$

Divide both sides by 2.

$$\frac{2m^{2}+3m}{2}=-\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\frac{3}{2}m=-\frac{1}{2}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{3}{2}m+\left(\frac{3}{4}\right)^{2}=-\frac{1}{2}+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}$$

Add -\frac{1}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=\frac{1}{16}$$

Factor m^{2}+\frac{3}{2}m+\frac{9}{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}.

$$\left(m+\frac{3}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$m+\frac{3}{4}=\frac{1}{4} m+\frac{3}{4}=-\frac{1}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$m=-\frac{1}{2} m=-1$$