Solve (-01-01)^2+(200-y)-(-115+4)^2=18225

$$( - 01 - 01 ) ^ { 2 } + \{ ( 200 - y ) - ( - 115 + 4 ) \} ^ { 2 } = 18225$$

Solve for y

$$y=176$$

$$y=446$$

Steps Using the Quadratic Formula

Multiply 0 and 1 to get 0.

$$\left(0-0\times 1\right)^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Multiply 0 and 1 to get 0.

$$\left(0-0\right)^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Subtracting 0 from itself leaves 0.

$$0^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Calculate 0 to the power of 2 and get 0.

$$0+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Add -115 and 4 to get -111.

$$0+\left(200-y-\left(-111\right)\right)^{2}=18225$$

The opposite of -111 is 111.

$$0+\left(200-y+111\right)^{2}=18225$$

Square 200-y+111.

$$0+y^{2}-622y+96721=18225$$

Add 0 and 96721 to get 96721.

$$96721+y^{2}-622y=18225$$

Subtract 18225 from both sides.

$$96721+y^{2}-622y-18225=0$$

Subtract 18225 from 96721 to get 78496.

$$78496+y^{2}-622y=0$$

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.

$$y^{2}-622y+78496=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -622 for b, and 78496 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-622\right)±\sqrt{\left(-622\right)^{2}-4\times 78496}}{2}$$

Square -622.

$$y=\frac{-\left(-622\right)±\sqrt{386884-4\times 78496}}{2}$$

Multiply -4 times 78496.

$$y=\frac{-\left(-622\right)±\sqrt{386884-313984}}{2}$$

Add 386884 to -313984.

$$y=\frac{-\left(-622\right)±\sqrt{72900}}{2}$$

Take the square root of 72900.

$$y=\frac{-\left(-622\right)±270}{2}$$

The opposite of -622 is 622.

$$y=\frac{622±270}{2}$$

Now solve the equation y=\frac{622±270}{2} when ± is plus. Add 622 to 270.

$$y=\frac{892}{2}$$

Divide 892 by 2.

$$y=446$$

Now solve the equation y=\frac{622±270}{2} when ± is minus. Subtract 270 from 622.

$$y=\frac{352}{2}$$

Divide 352 by 2.

$$y=176$$

The equation is now solved.

$$y=446 y=176$$

Steps for Completing the Square

Multiply 0 and 1 to get 0.

$$\left(0-0\times 1\right)^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Multiply 0 and 1 to get 0.

$$\left(0-0\right)^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Subtracting 0 from itself leaves 0.

$$0^{2}+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Calculate 0 to the power of 2 and get 0.

$$0+\left(200-y-\left(-115+4\right)\right)^{2}=18225$$

Add -115 and 4 to get -111.

$$0+\left(200-y-\left(-111\right)\right)^{2}=18225$$

The opposite of -111 is 111.

$$0+\left(200-y+111\right)^{2}=18225$$

Square 200-y+111.

$$0+y^{2}-622y+96721=18225$$

Add 0 and 96721 to get 96721.

$$96721+y^{2}-622y=18225$$

Subtract 96721 from both sides.

$$y^{2}-622y=18225-96721$$

Subtract 96721 from 18225 to get -78496.

$$y^{2}-622y=-78496$$

Divide -622, the coefficient of the x term, by 2 to get -311. Then add the square of -311 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-622y+\left(-311\right)^{2}=-78496+\left(-311\right)^{2}$$

Square -311.

$$y^{2}-622y+96721=-78496+96721$$

Add -78496 to 96721.

$$y^{2}-622y+96721=18225$$

Factor y^{2}-622y+96721. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-311\right)^{2}=18225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-311\right)^{2}}=\sqrt{18225}$$

Simplify.

$$y-311=135 y-311=-135$$

Add 311 to both sides of the equation.

$$y=446 y=176$$