Solve (x-27/2)^2=(18-x)^2+(24-45/2)^2

$$( x - \frac { 27 } { 2 } ) ^ { 2 } = ( 18 - x ) ^ { 2 } + ( 24 - \frac { 45 } { 2 } ) ^ { 2 }$$

Solve for x

$$x=16$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{27}{2}\right)^{2}.

$$x^{2}-27x+\frac{729}{4}=\left(18-x\right)^{2}+\left(24-\frac{45}{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(18-x\right)^{2}.

$$x^{2}-27x+\frac{729}{4}=324-36x+x^{2}+\left(24-\frac{45}{2}\right)^{2}$$

Subtract \frac{45}{2} from 24 to get \frac{3}{2}.

$$x^{2}-27x+\frac{729}{4}=324-36x+x^{2}+\left(\frac{3}{2}\right)^{2}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$x^{2}-27x+\frac{729}{4}=324-36x+x^{2}+\frac{9}{4}$$

Add 324 and \frac{9}{4} to get \frac{1305}{4}.

$$x^{2}-27x+\frac{729}{4}=\frac{1305}{4}-36x+x^{2}$$

Add 36x to both sides.

$$x^{2}-27x+\frac{729}{4}+36x=\frac{1305}{4}+x^{2}$$

Combine -27x and 36x to get 9x.

$$x^{2}+9x+\frac{729}{4}=\frac{1305}{4}+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+9x+\frac{729}{4}-x^{2}=\frac{1305}{4}$$

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

$$9x+\frac{729}{4}=\frac{1305}{4}$$

Subtract \frac{729}{4} from both sides.

$$9x=\frac{1305}{4}-\frac{729}{4}$$

Subtract \frac{729}{4} from \frac{1305}{4} to get 144.

$$9x=144$$

Divide both sides by 9.

$$x=\frac{144}{9}$$

Divide 144 by 9 to get 16.

$$x=16$$