Solve {left(12/5-7x/25right)}^2+{left(24x/25right)}^2=left(4-xright)^2

$${ \left( \frac{ 12 }{ 5 } - \frac{ 7x }{ 25 } \right) }^{ 2 } + { \left( \frac{ 24x }{ 25 } \right) }^{ 2 } = { \left(4-x \right) }^{ 2 }$$

Solve for x

$$x = \frac{20}{13} = 1\frac{7}{13} \approx 1.538461538$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 25 is 25. Multiply \frac{12}{5} times \frac{5}{5}.

$$\left(\frac{12\times 5}{25}-\frac{7x}{25}\right)^{2}+\left(\frac{24x}{25}\right)^{2}=\left(4-x\right)^{2}$$

Since \frac{12\times 5}{25} and \frac{7x}{25} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{12\times 5-7x}{25}\right)^{2}+\left(\frac{24x}{25}\right)^{2}=\left(4-x\right)^{2}$$

Do the multiplications in 12\times 5-7x.

$$\left(\frac{60-7x}{25}\right)^{2}+\left(\frac{24x}{25}\right)^{2}=\left(4-x\right)^{2}$$

To raise \frac{60-7x}{25} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(60-7x\right)^{2}}{25^{2}}+\left(\frac{24x}{25}\right)^{2}=\left(4-x\right)^{2}$$

To raise \frac{24x}{25} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(60-7x\right)^{2}}{25^{2}}+\frac{\left(24x\right)^{2}}{25^{2}}=\left(4-x\right)^{2}$$

Since \frac{\left(60-7x\right)^{2}}{25^{2}} and \frac{\left(24x\right)^{2}}{25^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(60-7x\right)^{2}+\left(24x\right)^{2}}{25^{2}}=\left(4-x\right)^{2}$$

Do the multiplications in \left(60-7x\right)^{2}+\left(24x\right)^{2}.

$$\frac{3600-840x+49x^{2}+\left(24x\right)^{2}}{25^{2}}=\left(4-x\right)^{2}$$

Combine like terms in 3600-840x+49x^{2}+\left(24x\right)^{2}.

$$\frac{3600-840x+625x^{2}}{25^{2}}=\left(4-x\right)^{2}$$

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

$$\frac{3600-840x+625x^{2}}{25^{2}}=16-8x+x^{2}$$

Calculate 25 to the power of 2 and get 625.

$$\frac{3600-840x+625x^{2}}{625}=16-8x+x^{2}$$

Divide each term of 3600-840x+625x^{2} by 625 to get \frac{144}{25}-\frac{168}{125}x+x^{2}.

$$\frac{144}{25}-\frac{168}{125}x+x^{2}=16-8x+x^{2}$$

Add 8x to both sides.

$$\frac{144}{25}-\frac{168}{125}x+x^{2}+8x=16+x^{2}$$

Combine -\frac{168}{125}x and 8x to get \frac{832}{125}x.

$$\frac{144}{25}+\frac{832}{125}x+x^{2}=16+x^{2}$$

Subtract x^{2} from both sides.

$$\frac{144}{25}+\frac{832}{125}x+x^{2}-x^{2}=16$$

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

$$\frac{144}{25}+\frac{832}{125}x=16$$

Subtract \frac{144}{25} from both sides.

$$\frac{832}{125}x=16-\frac{144}{25}$$

Subtract \frac{144}{25} from 16 to get \frac{256}{25}.

$$\frac{832}{125}x=\frac{256}{25}$$

Multiply both sides by \frac{125}{832}, the reciprocal of \frac{832}{125}.

$$x=\frac{256}{25}\times \frac{125}{832}$$

Multiply \frac{256}{25} and \frac{125}{832} to get \frac{20}{13}.

$$x=\frac{20}{13}$$