Solve 2(1/8m^2-1/4m-3)=6-m

$$2 ( \frac { 1 } { 8 } m ^ { 2 } - \frac { 1 } { 4 } m - 3 ) = 6 - m$$

Solve for m

$$m=-8$$

$$m=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by \frac{1}{8}m^{2}-\frac{1}{4}m-3.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-6=6-m$$

Subtract 6 from both sides.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-6-6=-m$$

Subtract 6 from -6 to get -12.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-12=-m$$

Add m to both sides.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-12+m=0$$

Combine -\frac{1}{2}m and m to get \frac{1}{2}m.

$$\frac{1}{4}m^{2}+\frac{1}{2}m-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, \frac{1}{2} for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\times \frac{1}{4}\left(-12\right)}}{2\times \frac{1}{4}}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$m=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\times \frac{1}{4}\left(-12\right)}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$m=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-\left(-12\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -12.

$$m=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+12}}{2\times \frac{1}{4}}$$

Add \frac{1}{4} to 12.

$$m=\frac{-\frac{1}{2}±\sqrt{\frac{49}{4}}}{2\times \frac{1}{4}}$$

Take the square root of \frac{49}{4}.

$$m=\frac{-\frac{1}{2}±\frac{7}{2}}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$m=\frac{-\frac{1}{2}±\frac{7}{2}}{\frac{1}{2}}$$

Now solve the equation m=\frac{-\frac{1}{2}±\frac{7}{2}}{\frac{1}{2}} when ± is plus. Add -\frac{1}{2} to \frac{7}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{3}{\frac{1}{2}}$$

Divide 3 by \frac{1}{2} by multiplying 3 by the reciprocal of \frac{1}{2}.

$$m=6$$

Now solve the equation m=\frac{-\frac{1}{2}±\frac{7}{2}}{\frac{1}{2}} when ± is minus. Subtract \frac{7}{2} from -\frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$m=-\frac{4}{\frac{1}{2}}$$

Divide -4 by \frac{1}{2} by multiplying -4 by the reciprocal of \frac{1}{2}.

$$m=-8$$

The equation is now solved.

$$m=6 m=-8$$

Steps for Completing the Square

Use the distributive property to multiply 2 by \frac{1}{8}m^{2}-\frac{1}{4}m-3.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-6=6-m$$

Add m to both sides.

$$\frac{1}{4}m^{2}-\frac{1}{2}m-6+m=6$$

Combine -\frac{1}{2}m and m to get \frac{1}{2}m.

$$\frac{1}{4}m^{2}+\frac{1}{2}m-6=6$$

Add 6 to both sides.

$$\frac{1}{4}m^{2}+\frac{1}{2}m=6+6$$

Add 6 and 6 to get 12.

$$\frac{1}{4}m^{2}+\frac{1}{2}m=12$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}m^{2}+\frac{1}{2}m}{\frac{1}{4}}=\frac{12}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$m^{2}+\frac{\frac{1}{2}}{\frac{1}{4}}m=\frac{12}{\frac{1}{4}}$$

Divide \frac{1}{2} by \frac{1}{4} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{4}.

$$m^{2}+2m=\frac{12}{\frac{1}{4}}$$

Divide 12 by \frac{1}{4} by multiplying 12 by the reciprocal of \frac{1}{4}.

$$m^{2}+2m=48$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+2m+1^{2}=48+1^{2}$$

Square 1.

$$m^{2}+2m+1=48+1$$

Add 48 to 1.

$$m^{2}+2m+1=49$$

Factor m^{2}+2m+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+1\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+1\right)^{2}}=\sqrt{49}$$

Simplify.

$$m+1=7 m+1=-7$$

Subtract 1 from both sides of the equation.

$$m=6 m=-8$$