Solve m^2-4m+8+m^2/2-6m+20=m^2/2-2m+4

$$m ^ { 2 } - 4 m + 8 + \frac { m ^ { 2 } } { 2 } - 6 m + 20 = \frac { m ^ { 2 } } { 2 } - 2 m + 4$$

Solve for m

$$m=4+2\sqrt{2}i\approx 4+2.828427125i$$

$$m=-2\sqrt{2}i+4\approx 4-2.828427125i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2m^{2}-8m+16+m^{2}-12m+40=m^{2}-4m+8$$

Combine 2m^{2} and m^{2} to get 3m^{2}.

$$3m^{2}-8m+16-12m+40=m^{2}-4m+8$$

Combine -8m and -12m to get -20m.

$$3m^{2}-20m+16+40=m^{2}-4m+8$$

Add 16 and 40 to get 56.

$$3m^{2}-20m+56=m^{2}-4m+8$$

Subtract m^{2} from both sides.

$$3m^{2}-20m+56-m^{2}=-4m+8$$

Combine 3m^{2} and -m^{2} to get 2m^{2}.

$$2m^{2}-20m+56=-4m+8$$

Add 4m to both sides.

$$2m^{2}-20m+56+4m=8$$

Combine -20m and 4m to get -16m.

$$2m^{2}-16m+56=8$$

Subtract 8 from both sides.

$$2m^{2}-16m+56-8=0$$

Subtract 8 from 56 to get 48.

$$2m^{2}-16m+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -16 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\times 48}}{2\times 2}$$

Square -16.

$$m=\frac{-\left(-16\right)±\sqrt{256-4\times 2\times 48}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-16\right)±\sqrt{256-8\times 48}}{2\times 2}$$

Multiply -8 times 48.

$$m=\frac{-\left(-16\right)±\sqrt{256-384}}{2\times 2}$$

Add 256 to -384.

$$m=\frac{-\left(-16\right)±\sqrt{-128}}{2\times 2}$$

Take the square root of -128.

$$m=\frac{-\left(-16\right)±8\sqrt{2}i}{2\times 2}$$

The opposite of -16 is 16.

$$m=\frac{16±8\sqrt{2}i}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{16±8\sqrt{2}i}{4}$$

Now solve the equation m=\frac{16±8\sqrt{2}i}{4} when ± is plus. Add 16 to 8i\sqrt{2}.

$$m=\frac{16+2^{\frac{7}{2}}i}{4}$$

Divide 16+i\times 2^{\frac{7}{2}} by 4.

$$m=4+2\sqrt{2}i$$

Now solve the equation m=\frac{16±8\sqrt{2}i}{4} when ± is minus. Subtract 8i\sqrt{2} from 16.

$$m=\frac{-2^{\frac{7}{2}}i+16}{4}$$

Divide 16-i\times 2^{\frac{7}{2}} by 4.

$$m=-2\sqrt{2}i+4$$

The equation is now solved.

$$m=4+2\sqrt{2}i m=-2\sqrt{2}i+4$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2m^{2}-8m+16+m^{2}-12m+40=m^{2}-4m+8$$

Combine 2m^{2} and m^{2} to get 3m^{2}.

$$3m^{2}-8m+16-12m+40=m^{2}-4m+8$$

Combine -8m and -12m to get -20m.

$$3m^{2}-20m+16+40=m^{2}-4m+8$$

Add 16 and 40 to get 56.

$$3m^{2}-20m+56=m^{2}-4m+8$$

Subtract m^{2} from both sides.

$$3m^{2}-20m+56-m^{2}=-4m+8$$

Combine 3m^{2} and -m^{2} to get 2m^{2}.

$$2m^{2}-20m+56=-4m+8$$

Add 4m to both sides.

$$2m^{2}-20m+56+4m=8$$

Combine -20m and 4m to get -16m.

$$2m^{2}-16m+56=8$$

Subtract 56 from both sides.

$$2m^{2}-16m=8-56$$

Subtract 56 from 8 to get -48.

$$2m^{2}-16m=-48$$

Divide both sides by 2.

$$\frac{2m^{2}-16m}{2}=-\frac{48}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\left(-\frac{16}{2}\right)m=-\frac{48}{2}$$

Divide -16 by 2.

$$m^{2}-8m=-\frac{48}{2}$$

Divide -48 by 2.

$$m^{2}-8m=-24$$

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.

$$m^{2}-8m+\left(-4\right)^{2}=-24+\left(-4\right)^{2}$$

Square -4.

$$m^{2}-8m+16=-24+16$$

Add -24 to 16.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m-4=2\sqrt{2}i m-4=-2\sqrt{2}i$$

Add 4 to both sides of the equation.

$$m=4+2\sqrt{2}i m=-2\sqrt{2}i+4$$