Solve 3m^{2}+(m/2+x)^{2}+(n-x-m^2/2)^2+3m^2=n^2

$$3 m ^ { 2 } + ( \frac { m } { 2 } + x ) ^ { 2 } + ( n - x - \frac { m ^ { 2 } } { 2 } ) ^ { 2 } + 3 m ^ { 2 } = n ^ { 2 }$$

Solve for n

$$\left\{\begin{matrix}n=\frac{8x^{2}+4xm^{2}+4mx+m^{4}+25m^{2}}{4\left(2x+m^{2}\right)}\text{, }&x\neq -\frac{m^{2}}{2}\\n\in \mathrm{R}\text{, }&m=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$3m^{2}+\left(\frac{m}{2}\right)^{2}+2\times \frac{m}{2}x+x^{2}+\left(n-x-\frac{m^{2}}{2}\right)^{2}+3m^{2}=n^{2}$$

To raise \frac{m}{2} to a power, raise both numerator and denominator to the power and then divide.

$$3m^{2}+\frac{m^{2}}{2^{2}}+2\times \frac{m}{2}x+x^{2}+\left(n-x-\frac{m^{2}}{2}\right)^{2}+3m^{2}=n^{2}$$

Express 2\times \frac{m}{2} as a single fraction.

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

Cancel out 2 and 2.

$$3m^{2}+\frac{m^{2}}{2^{2}}+mx+x^{2}+\left(n-x-\frac{m^{2}}{2}\right)^{2}+3m^{2}=n^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply mx+x^{2} times \frac{2^{2}}{2^{2}}.

$$3m^{2}+\frac{m^{2}}{2^{2}}+\frac{\left(mx+x^{2}\right)\times 2^{2}}{2^{2}}+\left(n-x-\frac{m^{2}}{2}\right)^{2}+3m^{2}=n^{2}$$

Since \frac{m^{2}}{2^{2}} and \frac{\left(mx+x^{2}\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$3m^{2}+\frac{m^{2}+\left(mx+x^{2}\right)\times 2^{2}}{2^{2}}+\left(n-x-\frac{m^{2}}{2}\right)^{2}+3m^{2}=n^{2}$$

Do the multiplications in m^{2}+\left(mx+x^{2}\right)\times 2^{2}.

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

Square n-x-\frac{m^{2}}{2}.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(-\frac{m^{2}}{2}\right)^{2}+2n\left(-\frac{m^{2}}{2}\right)-2x\left(-\frac{m^{2}}{2}\right)-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Calculate -\frac{m^{2}}{2} to the power of 2 and get \left(\frac{m^{2}}{2}\right)^{2}.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}+2n\left(-\frac{m^{2}}{2}\right)-2x\left(-\frac{m^{2}}{2}\right)-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Express 2\left(-\frac{m^{2}}{2}\right) as a single fraction.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}+\frac{-2m^{2}}{2}n-2x\left(-\frac{m^{2}}{2}\right)-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Cancel out 2 and 2.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n-2x\left(-\frac{m^{2}}{2}\right)-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Multiply -2 and -1 to get 2.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+2x\times \frac{m^{2}}{2}-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Express x\times \frac{m^{2}}{2} as a single fraction.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+2\times \frac{xm^{2}}{2}-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Express 2\times \frac{xm^{2}}{2} as a single fraction.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+\frac{2xm^{2}}{2}-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Cancel out 2 and 2.

$$3m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}+3m^{2}=n^{2}$$

Combine 3m^{2} and 3m^{2} to get 6m^{2}.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{2^{2}}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}=n^{2}$$

Calculate 2 to the power of 2 and get 4.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\left(\frac{m^{2}}{2}\right)^{2}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}=n^{2}$$

To raise \frac{m^{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\frac{\left(m^{2}\right)^{2}}{2^{2}}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}=n^{2}$$

Subtract n^{2} from both sides.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\frac{\left(m^{2}\right)^{2}}{2^{2}}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}-n^{2}=0$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\frac{m^{4}}{2^{2}}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}-n^{2}=0$$

Calculate 2 to the power of 2 and get 4.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\frac{m^{4}}{4}-m^{2}n+xm^{2}-2nx+n^{2}+x^{2}-n^{2}=0$$

Combine n^{2} and -n^{2} to get 0.

$$6m^{2}+\frac{m^{2}+4mx+4x^{2}}{4}+\frac{m^{4}}{4}-m^{2}n+xm^{2}-2nx+x^{2}=0$$

Subtract 6m^{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{m^{2}+4mx+4x^{2}}{4}+\frac{m^{4}}{4}-m^{2}n+xm^{2}-2nx+x^{2}=-6m^{2}$$

Subtract \frac{m^{2}+4mx+4x^{2}}{4} from both sides.

$$\frac{m^{4}}{4}-m^{2}n+xm^{2}-2nx+x^{2}=-6m^{2}-\frac{m^{2}+4mx+4x^{2}}{4}$$

Subtract \frac{m^{4}}{4} from both sides.

$$-m^{2}n+xm^{2}-2nx+x^{2}=-6m^{2}-\frac{m^{2}+4mx+4x^{2}}{4}-\frac{m^{4}}{4}$$

Since -\frac{m^{2}+4mx+4x^{2}}{4} and \frac{m^{4}}{4} have the same denominator, subtract them by subtracting their numerators.

$$-m^{2}n+xm^{2}-2nx+x^{2}=-6m^{2}+\frac{-\left(m^{2}+4mx+4x^{2}\right)-m^{4}}{4}$$

Do the multiplications in -\left(m^{2}+4mx+4x^{2}\right)-m^{4}.

$$-m^{2}n+xm^{2}-2nx+x^{2}=-6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}$$

Subtract xm^{2} from both sides.

$$-m^{2}n-2nx+x^{2}=-6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-xm^{2}$$

Subtract x^{2} from both sides.

$$-m^{2}n-2nx=-6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-xm^{2}-x^{2}$$

Multiply both sides of the equation by 4.

$$-4m^{2}n-8nx=4\left(-6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-xm^{2}\right)-4x^{2}$$

Multiply both sides of the equation by 4.

$$-16m^{2}n-32nx=16\left(-6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-xm^{2}\right)-16x^{2}$$

Use the distributive property to multiply 16 by -6m^{2}+\frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-xm^{2}.

$$-16m^{2}n-32nx=-96m^{2}+16\times \frac{-m^{2}-4mx-4x^{2}-m^{4}}{4}-16xm^{2}-16x^{2}$$

Cancel out 4, the greatest common factor in 16 and 4.

$$-16m^{2}n-32nx=-96m^{2}+4\left(-m^{2}-4mx-4x^{2}-m^{4}\right)-16xm^{2}-16x^{2}$$

Use the distributive property to multiply 4 by -m^{2}-4mx-4x^{2}-m^{4}.

$$-16m^{2}n-32nx=-96m^{2}-4m^{2}-16mx-16x^{2}-4m^{4}-16xm^{2}-16x^{2}$$

Combine -96m^{2} and -4m^{2} to get -100m^{2}.

$$-16m^{2}n-32nx=-100m^{2}-16mx-16x^{2}-4m^{4}-16xm^{2}-16x^{2}$$

Combine -16x^{2} and -16x^{2} to get -32x^{2}.

$$-16m^{2}n-32nx=-100m^{2}-16mx-32x^{2}-4m^{4}-16xm^{2}$$

Combine all terms containing n.

$$\left(-16m^{2}-32x\right)n=-100m^{2}-16mx-32x^{2}-4m^{4}-16xm^{2}$$

The equation is in standard form.

$$\left(-32x-16m^{2}\right)n=-32x^{2}-16xm^{2}-16mx-4m^{4}-100m^{2}$$

Divide both sides by -16m^{2}-32x.

$$\frac{\left(-32x-16m^{2}\right)n}{-32x-16m^{2}}=\frac{-32x^{2}-16xm^{2}-16mx-4m^{4}-100m^{2}}{-32x-16m^{2}}$$

Dividing by -16m^{2}-32x undoes the multiplication by -16m^{2}-32x.

$$n=\frac{-32x^{2}-16xm^{2}-16mx-4m^{4}-100m^{2}}{-32x-16m^{2}}$$

Divide -100m^{2}-16mx-32x^{2}-4m^{4}-16xm^{2} by -16m^{2}-32x.

$$n=\frac{8x^{2}+4xm^{2}+4mx+m^{4}+25m^{2}}{4\left(2x+m^{2}\right)}$$