Solve {left(m-3/m+5right)}^2+{left(4-2m/m+5right)}^2=1

$${ \left( \frac{ m-3 }{ m+5 } \right) }^{ 2 } + { \left( \frac{ 4-2m }{ m+5 } \right) }^{ 2 } =1$$

Solve for m

$$m=8$$

$$m=0$$

Steps Using Factoring

To raise \frac{m-3}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\left(\frac{4-2m}{m+5}\right)^{2}=1$$

To raise \frac{4-2m}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Since \frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}} and \frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(m-3\right)^{2}+\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Do the multiplications in \left(m-3\right)^{2}+\left(4-2m\right)^{2}.

$$\frac{m^{2}-6m+9+16-16m+4m^{2}}{\left(m+5\right)^{2}}=1$$

Combine like terms in m^{2}-6m+9+16-16m+4m^{2}.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+5\right)^{2}.

$$\frac{5m^{2}-22m+25}{m^{2}+10m+25}=1$$

Subtract 1 from both sides.

$$\frac{5m^{2}-22m+25}{m^{2}+10m+25}-1=0$$

Factor m^{2}+10m+25.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}-1=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}}.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}-\frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}}=0$$

Since \frac{5m^{2}-22m+25}{\left(m+5\right)^{2}} and \frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5m^{2}-22m+25-\left(m+5\right)^{2}}{\left(m+5\right)^{2}}=0$$

Do the multiplications in 5m^{2}-22m+25-\left(m+5\right)^{2}.

$$\frac{5m^{2}-22m+25-m^{2}-10m-25}{\left(m+5\right)^{2}}=0$$

Combine like terms in 5m^{2}-22m+25-m^{2}-10m-25.

$$\frac{4m^{2}-32m}{\left(m+5\right)^{2}}=0$$

Variable m cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by \left(m+5\right)^{2}.

$$4m^{2}-32m=0$$

Factor out m.

$$m\left(4m-32\right)=0$$

To find equation solutions, solve m=0 and 4m-32=0.

$$m=0 m=8$$

Steps Using the Quadratic Formula

To raise \frac{m-3}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\left(\frac{4-2m}{m+5}\right)^{2}=1$$

To raise \frac{4-2m}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Since \frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}} and \frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(m-3\right)^{2}+\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Do the multiplications in \left(m-3\right)^{2}+\left(4-2m\right)^{2}.

$$\frac{m^{2}-6m+9+16-16m+4m^{2}}{\left(m+5\right)^{2}}=1$$

Combine like terms in m^{2}-6m+9+16-16m+4m^{2}.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+5\right)^{2}.

$$\frac{5m^{2}-22m+25}{m^{2}+10m+25}=1$$

Subtract 1 from both sides.

$$\frac{5m^{2}-22m+25}{m^{2}+10m+25}-1=0$$

Factor m^{2}+10m+25.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}-1=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}}.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}-\frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}}=0$$

Since \frac{5m^{2}-22m+25}{\left(m+5\right)^{2}} and \frac{\left(m+5\right)^{2}}{\left(m+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5m^{2}-22m+25-\left(m+5\right)^{2}}{\left(m+5\right)^{2}}=0$$

Do the multiplications in 5m^{2}-22m+25-\left(m+5\right)^{2}.

$$\frac{5m^{2}-22m+25-m^{2}-10m-25}{\left(m+5\right)^{2}}=0$$

Combine like terms in 5m^{2}-22m+25-m^{2}-10m-25.

$$\frac{4m^{2}-32m}{\left(m+5\right)^{2}}=0$$

Variable m cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by \left(m+5\right)^{2}.

$$4m^{2}-32m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -32 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}}}{2\times 4}$$

Take the square root of \left(-32\right)^{2}.

$$m=\frac{-\left(-32\right)±32}{2\times 4}$$

The opposite of -32 is 32.

$$m=\frac{32±32}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{32±32}{8}$$

Now solve the equation m=\frac{32±32}{8} when ± is plus. Add 32 to 32.

$$m=\frac{64}{8}$$

Divide 64 by 8.

$$m=8$$

Now solve the equation m=\frac{32±32}{8} when ± is minus. Subtract 32 from 32.

$$m=\frac{0}{8}$$

Divide 0 by 8.

$$m=0$$

The equation is now solved.

$$m=8 m=0$$

Steps for Completing the Square

To raise \frac{m-3}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\left(\frac{4-2m}{m+5}\right)^{2}=1$$

To raise \frac{4-2m}{m+5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}}+\frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Since \frac{\left(m-3\right)^{2}}{\left(m+5\right)^{2}} and \frac{\left(4-2m\right)^{2}}{\left(m+5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(m-3\right)^{2}+\left(4-2m\right)^{2}}{\left(m+5\right)^{2}}=1$$

Do the multiplications in \left(m-3\right)^{2}+\left(4-2m\right)^{2}.

$$\frac{m^{2}-6m+9+16-16m+4m^{2}}{\left(m+5\right)^{2}}=1$$

Combine like terms in m^{2}-6m+9+16-16m+4m^{2}.

$$\frac{5m^{2}-22m+25}{\left(m+5\right)^{2}}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+5\right)^{2}.

$$\frac{5m^{2}-22m+25}{m^{2}+10m+25}=1$$

Variable m cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by \left(m+5\right)^{2}.

$$5m^{2}-22m+25=\left(m+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+5\right)^{2}.

$$5m^{2}-22m+25=m^{2}+10m+25$$

Subtract m^{2} from both sides.

$$5m^{2}-22m+25-m^{2}=10m+25$$

Combine 5m^{2} and -m^{2} to get 4m^{2}.

$$4m^{2}-22m+25=10m+25$$

Subtract 10m from both sides.

$$4m^{2}-22m+25-10m=25$$

Combine -22m and -10m to get -32m.

$$4m^{2}-32m+25=25$$

Subtract 25 from both sides.

$$4m^{2}-32m=25-25$$

Subtract 25 from 25 to get 0.

$$4m^{2}-32m=0$$

Divide both sides by 4.

$$\frac{4m^{2}-32m}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}+\left(-\frac{32}{4}\right)m=\frac{0}{4}$$

Divide -32 by 4.

$$m^{2}-8m=\frac{0}{4}$$

Divide 0 by 4.

$$m^{2}-8m=0$$

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.

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

Square -4.

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

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}.

$$\left(m-4\right)^{2}=16$$

Take the square root of both sides of the equation.

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

Simplify.

$$m-4=4 m-4=-4$$

Add 4 to both sides of the equation.

$$m=8 m=0$$