Solve (4-x)^2+(-1/2x+2)^2=(5-x)^2+(-1/2x+2)^2

$$( 4 - x ) ^ { 2 } + ( - \frac { 1 } { 2 } x + 2 ) ^ { 2 } = ( 5 - x ) ^ { 2 } + ( - \frac { 1 } { 2 } x + 2 ) ^ { 2 }$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps for Solving Linear Equation

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

$$16-8x+x^{2}+\left(-\frac{1}{2}x+2\right)^{2}=\left(5-x\right)^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{2}x+2\right)^{2}.

$$16-8x+x^{2}+\frac{1}{4}x^{2}-2x+4=\left(5-x\right)^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

Combine x^{2} and \frac{1}{4}x^{2} to get \frac{5}{4}x^{2}.

$$16-8x+\frac{5}{4}x^{2}-2x+4=\left(5-x\right)^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

Combine -8x and -2x to get -10x.

$$16-10x+\frac{5}{4}x^{2}+4=\left(5-x\right)^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

Add 16 and 4 to get 20.

$$20-10x+\frac{5}{4}x^{2}=\left(5-x\right)^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

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

$$20-10x+\frac{5}{4}x^{2}=25-10x+x^{2}+\left(-\frac{1}{2}x+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{2}x+2\right)^{2}.

$$20-10x+\frac{5}{4}x^{2}=25-10x+x^{2}+\frac{1}{4}x^{2}-2x+4$$

Combine x^{2} and \frac{1}{4}x^{2} to get \frac{5}{4}x^{2}.

$$20-10x+\frac{5}{4}x^{2}=25-10x+\frac{5}{4}x^{2}-2x+4$$

Combine -10x and -2x to get -12x.

$$20-10x+\frac{5}{4}x^{2}=25-12x+\frac{5}{4}x^{2}+4$$

Add 25 and 4 to get 29.

$$20-10x+\frac{5}{4}x^{2}=29-12x+\frac{5}{4}x^{2}$$

Add 12x to both sides.

$$20-10x+\frac{5}{4}x^{2}+12x=29+\frac{5}{4}x^{2}$$

Combine -10x and 12x to get 2x.

$$20+2x+\frac{5}{4}x^{2}=29+\frac{5}{4}x^{2}$$

Subtract \frac{5}{4}x^{2} from both sides.

$$20+2x+\frac{5}{4}x^{2}-\frac{5}{4}x^{2}=29$$

Combine \frac{5}{4}x^{2} and -\frac{5}{4}x^{2} to get 0.

$$20+2x=29$$

Subtract 20 from both sides.

$$2x=29-20$$

Subtract 20 from 29 to get 9.

$$2x=9$$

Divide both sides by 2.

$$x=\frac{9}{2}$$