Solve (7-z)^2+17=(z+2)^2+34

$$( 7 - z ) ^ { 2 } + 17 = ( z + 2 ) ^ { 2 } + 34$$

Solve for z

$$z = \frac{14}{9} = 1\frac{5}{9} \approx 1.555555556$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-z\right)^{2}.

$$49-14z+z^{2}+17=\left(z+2\right)^{2}+34$$

Add 49 and 17 to get 66.

$$66-14z+z^{2}=\left(z+2\right)^{2}+34$$

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

$$66-14z+z^{2}=z^{2}+4z+4+34$$

Add 4 and 34 to get 38.

$$66-14z+z^{2}=z^{2}+4z+38$$

Subtract z^{2} from both sides.

$$66-14z+z^{2}-z^{2}=4z+38$$

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

$$66-14z=4z+38$$

Subtract 4z from both sides.

$$66-14z-4z=38$$

Combine -14z and -4z to get -18z.

$$66-18z=38$$

Subtract 66 from both sides.

$$-18z=38-66$$

Subtract 66 from 38 to get -28.

$$-18z=-28$$

Divide both sides by -18.

$$z=\frac{-28}{-18}$$

Reduce the fraction \frac{-28}{-18} to lowest terms by extracting and canceling out -2.

$$z=\frac{14}{9}$$