Solve (-0.1-0.1)^2+(2.00-y)-(-11.5+4)^2=182.25

$$( - 0.1 - 0.1 ) ^ { 2 } + \{ ( 2.00 - y ) - ( - 11.5 + 4 ) \} ^ { 2 } = 182.25$$

Solve for y

$$y=-\frac{\sqrt{18221}}{10}+9.5\approx -3.998518437$$

$$y=\frac{\sqrt{18221}}{10}+9.5\approx 22.998518437$$

Steps Using the Quadratic Formula

Subtract 0.1 from -0.1 to get -0.2.

$$\left(-0.2\right)^{2}+\left(2-y-\left(-11.5+4\right)\right)^{2}=182.25$$

Calculate -0.2 to the power of 2 and get 0.04.

$$0.04+\left(2-y-\left(-11.5+4\right)\right)^{2}=182.25$$

Add -11.5 and 4 to get -7.5.

$$0.04+\left(2-y-\left(-7.5\right)\right)^{2}=182.25$$

The opposite of -7.5 is 7.5.

$$0.04+\left(2-y+7.5\right)^{2}=182.25$$

Square 2-y+7.5.

$$0.04+y^{2}-19y+90.25=182.25$$

Add 0.04 and 90.25 to get 90.29.

$$90.29+y^{2}-19y=182.25$$

Subtract 182.25 from both sides.

$$90.29+y^{2}-19y-182.25=0$$

Subtract 182.25 from 90.29 to get -91.96.

$$-91.96+y^{2}-19y=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y^{2}-19y-91.96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -19 for b, and -91.96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\left(-91.96\right)}}{2}$$

Square -19.

$$y=\frac{-\left(-19\right)±\sqrt{361-4\left(-91.96\right)}}{2}$$

Multiply -4 times -91.96.

$$y=\frac{-\left(-19\right)±\sqrt{361+367.84}}{2}$$

Add 361 to 367.84.

$$y=\frac{-\left(-19\right)±\sqrt{728.84}}{2}$$

Take the square root of 728.84.

$$y=\frac{-\left(-19\right)±\frac{\sqrt{18221}}{5}}{2}$$

The opposite of -19 is 19.

$$y=\frac{19±\frac{\sqrt{18221}}{5}}{2}$$

Now solve the equation y=\frac{19±\frac{\sqrt{18221}}{5}}{2} when ± is plus. Add 19 to \frac{\sqrt{18221}}{5}.

$$y=\frac{\frac{\sqrt{18221}}{5}+19}{2}$$

Divide 19+\frac{\sqrt{18221}}{5} by 2.

$$y=\frac{\sqrt{18221}}{10}+\frac{19}{2}$$

Now solve the equation y=\frac{19±\frac{\sqrt{18221}}{5}}{2} when ± is minus. Subtract \frac{\sqrt{18221}}{5} from 19.

$$y=\frac{-\frac{\sqrt{18221}}{5}+19}{2}$$

Divide 19-\frac{\sqrt{18221}}{5} by 2.

$$y=-\frac{\sqrt{18221}}{10}+\frac{19}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{18221}}{10}+\frac{19}{2} y=-\frac{\sqrt{18221}}{10}+\frac{19}{2}$$

Steps for Completing the Square

Subtract 0.1 from -0.1 to get -0.2.

$$\left(-0.2\right)^{2}+\left(2-y-\left(-11.5+4\right)\right)^{2}=182.25$$

Calculate -0.2 to the power of 2 and get 0.04.

$$0.04+\left(2-y-\left(-11.5+4\right)\right)^{2}=182.25$$

Add -11.5 and 4 to get -7.5.

$$0.04+\left(2-y-\left(-7.5\right)\right)^{2}=182.25$$

The opposite of -7.5 is 7.5.

$$0.04+\left(2-y+7.5\right)^{2}=182.25$$

Square 2-y+7.5.

$$0.04+y^{2}-19y+90.25=182.25$$

Add 0.04 and 90.25 to get 90.29.

$$90.29+y^{2}-19y=182.25$$

Subtract 90.29 from both sides.

$$y^{2}-19y=182.25-90.29$$

Subtract 90.29 from 182.25 to get 91.96.

$$y^{2}-19y=91.96$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-19y+\left(-\frac{19}{2}\right)^{2}=91.96+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-19y+\frac{361}{4}=91.96+\frac{361}{4}$$

Add 91.96 to \frac{361}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-19y+\frac{361}{4}=\frac{18221}{100}$$

Factor y^{2}-19y+\frac{361}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{19}{2}\right)^{2}=\frac{18221}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{19}{2}\right)^{2}}=\sqrt{\frac{18221}{100}}$$

Simplify.

$$y-\frac{19}{2}=\frac{\sqrt{18221}}{10} y-\frac{19}{2}=-\frac{\sqrt{18221}}{10}$$

Add \frac{19}{2} to both sides of the equation.

$$y=\frac{\sqrt{18221}}{10}+\frac{19}{2} y=-\frac{\sqrt{18221}}{10}+\frac{19}{2}$$