Solve 3left(m-2right)=-(1/2left(m-2right)^2-8)

$$3 \left( m-2 \right) = -( \frac{ 1 }{ 2 } { \left(m-2 \right) }^{ 2 } -8)$$

Solve for m

$$m=-6$$

$$m=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by m-2.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2\right)^{2}.

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

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

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

Subtract 8 from 2 to get -6.

$$3m-6=-\left(\frac{1}{2}m^{2}-2m-6\right)$$

To find the opposite of \frac{1}{2}m^{2}-2m-6, find the opposite of each term.

$$3m-6=-\frac{1}{2}m^{2}+2m+6$$

Add \frac{1}{2}m^{2} to both sides.

$$3m-6+\frac{1}{2}m^{2}=2m+6$$

Subtract 2m from both sides.

$$3m-6+\frac{1}{2}m^{2}-2m=6$$

Combine 3m and -2m to get m.

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

Subtract 6 from both sides.

$$m-6+\frac{1}{2}m^{2}-6=0$$

Subtract 6 from -6 to get -12.

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

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.

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

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

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

Square 1.

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

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

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

Multiply -2 times -12.

$$m=\frac{-1±\sqrt{1+24}}{2\times \frac{1}{2}}$$

Add 1 to 24.

$$m=\frac{-1±\sqrt{25}}{2\times \frac{1}{2}}$$

Take the square root of 25.

$$m=\frac{-1±5}{2\times \frac{1}{2}}$$

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

$$m=\frac{-1±5}{1}$$

Now solve the equation m=\frac{-1±5}{1} when ± is plus. Add -1 to 5.

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

Divide 4 by 1.

$$m=4$$

Now solve the equation m=\frac{-1±5}{1} when ± is minus. Subtract 5 from -1.

$$m=-\frac{6}{1}$$

Divide -6 by 1.

$$m=-6$$

The equation is now solved.

$$m=4 m=-6$$

Steps for Completing the Square

Use the distributive property to multiply 3 by m-2.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2\right)^{2}.

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

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

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

Subtract 8 from 2 to get -6.

$$3m-6=-\left(\frac{1}{2}m^{2}-2m-6\right)$$

To find the opposite of \frac{1}{2}m^{2}-2m-6, find the opposite of each term.

$$3m-6=-\frac{1}{2}m^{2}+2m+6$$

Add \frac{1}{2}m^{2} to both sides.

$$3m-6+\frac{1}{2}m^{2}=2m+6$$

Subtract 2m from both sides.

$$3m-6+\frac{1}{2}m^{2}-2m=6$$

Combine 3m and -2m to get m.

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

Add 6 to both sides.

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

Add 6 and 6 to get 12.

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

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.

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

Multiply both sides by 2.

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

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

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

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

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

Divide 12 by \frac{1}{2} by multiplying 12 by the reciprocal of \frac{1}{2}.

$$m^{2}+2m=24$$

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.

$$m^{2}+2m+1^{2}=24+1^{2}$$

Square 1.

$$m^{2}+2m+1=24+1$$

Add 24 to 1.

$$m^{2}+2m+1=25$$

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}.

$$\left(m+1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+1\right)^{2}}=\sqrt{25}$$

Simplify.

$$m+1=5 m+1=-5$$

Subtract 1 from both sides of the equation.

$$m=4 m=-6$$