Solve (m+2)^2+2(1/2m+2)^2+(4-m)^2=36

$$( m + 2 ) ^ { 2 } + 2 ( \frac { 1 } { 2 } m + 2 ) ^ { 2 } + ( 4 - m ) ^ { 2 } = 36$$

Solve for m

$$m = \frac{4 \sqrt{5}}{5} \approx 1.788854382$$

$$m = -\frac{4 \sqrt{5}}{5} \approx -1.788854382$$

Steps by Finding Square Root

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

$$m^{2}+4m+4+2\left(\frac{1}{2}m+2\right)^{2}+\left(4-m\right)^{2}=36$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{2}m+2\right)^{2}.

$$m^{2}+4m+4+2\left(\frac{1}{4}m^{2}+2m+4\right)+\left(4-m\right)^{2}=36$$

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

$$m^{2}+4m+4+\frac{1}{2}m^{2}+4m+8+\left(4-m\right)^{2}=36$$

Combine m^{2} and \frac{1}{2}m^{2} to get \frac{3}{2}m^{2}.

$$\frac{3}{2}m^{2}+4m+4+4m+8+\left(4-m\right)^{2}=36$$

Combine 4m and 4m to get 8m.

$$\frac{3}{2}m^{2}+8m+4+8+\left(4-m\right)^{2}=36$$

Add 4 and 8 to get 12.

$$\frac{3}{2}m^{2}+8m+12+\left(4-m\right)^{2}=36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-m\right)^{2}.

$$\frac{3}{2}m^{2}+8m+12+16-8m+m^{2}=36$$

Add 12 and 16 to get 28.

$$\frac{3}{2}m^{2}+8m+28-8m+m^{2}=36$$

Combine 8m and -8m to get 0.

$$\frac{3}{2}m^{2}+28+m^{2}=36$$

Combine \frac{3}{2}m^{2} and m^{2} to get \frac{5}{2}m^{2}.

$$\frac{5}{2}m^{2}+28=36$$

Subtract 28 from both sides.

$$\frac{5}{2}m^{2}=36-28$$

Subtract 28 from 36 to get 8.

$$\frac{5}{2}m^{2}=8$$

Multiply both sides by \frac{2}{5}, the reciprocal of \frac{5}{2}.

$$m^{2}=8\times \frac{2}{5}$$

Multiply 8 and \frac{2}{5} to get \frac{16}{5}.

$$m^{2}=\frac{16}{5}$$

Take the square root of both sides of the equation.

$$m=\frac{4\sqrt{5}}{5} m=-\frac{4\sqrt{5}}{5}$$

Steps Using the Quadratic Formula

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

$$m^{2}+4m+4+2\left(\frac{1}{2}m+2\right)^{2}+\left(4-m\right)^{2}=36$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{2}m+2\right)^{2}.

$$m^{2}+4m+4+2\left(\frac{1}{4}m^{2}+2m+4\right)+\left(4-m\right)^{2}=36$$

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

$$m^{2}+4m+4+\frac{1}{2}m^{2}+4m+8+\left(4-m\right)^{2}=36$$

Combine m^{2} and \frac{1}{2}m^{2} to get \frac{3}{2}m^{2}.

$$\frac{3}{2}m^{2}+4m+4+4m+8+\left(4-m\right)^{2}=36$$

Combine 4m and 4m to get 8m.

$$\frac{3}{2}m^{2}+8m+4+8+\left(4-m\right)^{2}=36$$

Add 4 and 8 to get 12.

$$\frac{3}{2}m^{2}+8m+12+\left(4-m\right)^{2}=36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-m\right)^{2}.

$$\frac{3}{2}m^{2}+8m+12+16-8m+m^{2}=36$$

Add 12 and 16 to get 28.

$$\frac{3}{2}m^{2}+8m+28-8m+m^{2}=36$$

Combine 8m and -8m to get 0.

$$\frac{3}{2}m^{2}+28+m^{2}=36$$

Combine \frac{3}{2}m^{2} and m^{2} to get \frac{5}{2}m^{2}.

$$\frac{5}{2}m^{2}+28=36$$

Subtract 36 from both sides.

$$\frac{5}{2}m^{2}+28-36=0$$

Subtract 36 from 28 to get -8.

$$\frac{5}{2}m^{2}-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{2} for a, 0 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{0±\sqrt{0^{2}-4\times \frac{5}{2}\left(-8\right)}}{2\times \frac{5}{2}}$$

Square 0.

$$m=\frac{0±\sqrt{-4\times \frac{5}{2}\left(-8\right)}}{2\times \frac{5}{2}}$$

Multiply -4 times \frac{5}{2}.

$$m=\frac{0±\sqrt{-10\left(-8\right)}}{2\times \frac{5}{2}}$$

Multiply -10 times -8.

$$m=\frac{0±\sqrt{80}}{2\times \frac{5}{2}}$$

Take the square root of 80.

$$m=\frac{0±4\sqrt{5}}{2\times \frac{5}{2}}$$

Multiply 2 times \frac{5}{2}.

$$m=\frac{0±4\sqrt{5}}{5}$$

Now solve the equation m=\frac{0±4\sqrt{5}}{5} when ± is plus.

$$m=\frac{4\sqrt{5}}{5}$$

Now solve the equation m=\frac{0±4\sqrt{5}}{5} when ± is minus.

$$m=-\frac{4\sqrt{5}}{5}$$

The equation is now solved.

$$m=\frac{4\sqrt{5}}{5} m=-\frac{4\sqrt{5}}{5}$$