Solve 9+{left(4-x/4right)}^2={left(8-x/4right)}^2-25

$$9+ { \left(4- \frac{ x }{ 4 } \right) }^{ 2 } = { \left(8- \frac{ x }{ 4 } \right) }^{ 2 } -25$$

Solve for x

$$x=7$$

Steps for Solving Linear Equation

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

$$9+16+8\left(-\frac{x}{4}\right)+\left(-\frac{x}{4}\right)^{2}=\left(8-\frac{x}{4}\right)^{2}-25$$

Cancel out 4, the greatest common factor in 8 and 4.

$$9+16-2x+\left(-\frac{x}{4}\right)^{2}=\left(8-\frac{x}{4}\right)^{2}-25$$

Calculate -\frac{x}{4} to the power of 2 and get \left(\frac{x}{4}\right)^{2}.

$$9+16-2x+\left(\frac{x}{4}\right)^{2}=\left(8-\frac{x}{4}\right)^{2}-25$$

Add 9 and 16 to get 25.

$$25-2x+\left(\frac{x}{4}\right)^{2}=\left(8-\frac{x}{4}\right)^{2}-25$$

To raise \frac{x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$25-2x+\frac{x^{2}}{4^{2}}=\left(8-\frac{x}{4}\right)^{2}-25$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25-2x times \frac{4^{2}}{4^{2}}.

$$\frac{\left(25-2x\right)\times 4^{2}}{4^{2}}+\frac{x^{2}}{4^{2}}=\left(8-\frac{x}{4}\right)^{2}-25$$

Since \frac{\left(25-2x\right)\times 4^{2}}{4^{2}} and \frac{x^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(25-2x\right)\times 4^{2}+x^{2}}{4^{2}}=\left(8-\frac{x}{4}\right)^{2}-25$$

Do the multiplications in \left(25-2x\right)\times 4^{2}+x^{2}.

$$\frac{400-32x+x^{2}}{4^{2}}=\left(8-\frac{x}{4}\right)^{2}-25$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8-\frac{x}{4}\right)^{2}.

$$\frac{400-32x+x^{2}}{4^{2}}=64+16\left(-\frac{x}{4}\right)+\left(-\frac{x}{4}\right)^{2}-25$$

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

$$\frac{400-32x+x^{2}}{4^{2}}=64-4x+\left(-\frac{x}{4}\right)^{2}-25$$

Calculate -\frac{x}{4} to the power of 2 and get \left(\frac{x}{4}\right)^{2}.

$$\frac{400-32x+x^{2}}{4^{2}}=64-4x+\left(\frac{x}{4}\right)^{2}-25$$

To raise \frac{x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{400-32x+x^{2}}{4^{2}}=64-4x+\frac{x^{2}}{4^{2}}-25$$

Subtract 25 from 64 to get 39.

$$\frac{400-32x+x^{2}}{4^{2}}=39-4x+\frac{x^{2}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{400-32x+x^{2}}{4^{2}}=39-4x+\frac{x^{2}}{16}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{400-32x+x^{2}}{16}=39-4x+\frac{x^{2}}{16}$$

Divide each term of 400-32x+x^{2} by 16 to get 25-2x+\frac{1}{16}x^{2}.

$$25-2x+\frac{1}{16}x^{2}=39-4x+\frac{x^{2}}{16}$$

Add 4x to both sides.

$$25-2x+\frac{1}{16}x^{2}+4x=39+\frac{x^{2}}{16}$$

Combine -2x and 4x to get 2x.

$$25+2x+\frac{1}{16}x^{2}=39+\frac{x^{2}}{16}$$

Subtract \frac{x^{2}}{16} from both sides.

$$25+2x+\frac{1}{16}x^{2}-\frac{x^{2}}{16}=39$$

Combine \frac{1}{16}x^{2} and -\frac{x^{2}}{16} to get 0.

$$25+2x=39$$

Subtract 25 from both sides.

$$2x=39-25$$

Subtract 25 from 39 to get 14.

$$2x=14$$

Divide both sides by 2.

$$x=\frac{14}{2}$$

Divide 14 by 2 to get 7.

$$x=7$$