Solve (y+202)^2+y^2+202y+y^2=10

$$( y + 202 ) ^ { 2 } + y ^ { 2 } + 202 y + y ^ { 2 } = 10$$

Solve for y

$$y=-101+\sqrt{3397}i\approx -101+58.283788484i$$

$$y=-\sqrt{3397}i-101\approx -101-58.283788484i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+202\right)^{2}.

$$y^{2}+404y+40804+y^{2}+202y+y^{2}=10$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$2y^{2}+404y+40804+202y+y^{2}=10$$

Combine 404y and 202y to get 606y.

$$2y^{2}+606y+40804+y^{2}=10$$

Combine 2y^{2} and y^{2} to get 3y^{2}.

$$3y^{2}+606y+40804=10$$

Subtract 10 from both sides.

$$3y^{2}+606y+40804-10=0$$

Subtract 10 from 40804 to get 40794.

$$3y^{2}+606y+40794=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 606 for b, and 40794 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-606±\sqrt{606^{2}-4\times 3\times 40794}}{2\times 3}$$

Square 606.

$$y=\frac{-606±\sqrt{367236-4\times 3\times 40794}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-606±\sqrt{367236-12\times 40794}}{2\times 3}$$

Multiply -12 times 40794.

$$y=\frac{-606±\sqrt{367236-489528}}{2\times 3}$$

Add 367236 to -489528.

$$y=\frac{-606±\sqrt{-122292}}{2\times 3}$$

Take the square root of -122292.

$$y=\frac{-606±6\sqrt{3397}i}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{-606±6\sqrt{3397}i}{6}$$

Now solve the equation y=\frac{-606±6\sqrt{3397}i}{6} when ± is plus. Add -606 to 6i\sqrt{3397}.

$$y=\frac{-606+6\sqrt{3397}i}{6}$$

Divide -606+6i\sqrt{3397} by 6.

$$y=-101+\sqrt{3397}i$$

Now solve the equation y=\frac{-606±6\sqrt{3397}i}{6} when ± is minus. Subtract 6i\sqrt{3397} from -606.

$$y=\frac{-6\sqrt{3397}i-606}{6}$$

Divide -606-6i\sqrt{3397} by 6.

$$y=-\sqrt{3397}i-101$$

The equation is now solved.

$$y=-101+\sqrt{3397}i y=-\sqrt{3397}i-101$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+202\right)^{2}.

$$y^{2}+404y+40804+y^{2}+202y+y^{2}=10$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$2y^{2}+404y+40804+202y+y^{2}=10$$

Combine 404y and 202y to get 606y.

$$2y^{2}+606y+40804+y^{2}=10$$

Combine 2y^{2} and y^{2} to get 3y^{2}.

$$3y^{2}+606y+40804=10$$

Subtract 40804 from both sides.

$$3y^{2}+606y=10-40804$$

Subtract 40804 from 10 to get -40794.

$$3y^{2}+606y=-40794$$

Divide both sides by 3.

$$\frac{3y^{2}+606y}{3}=-\frac{40794}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}+\frac{606}{3}y=-\frac{40794}{3}$$

Divide 606 by 3.

$$y^{2}+202y=-\frac{40794}{3}$$

Divide -40794 by 3.

$$y^{2}+202y=-13598$$

Divide 202, the coefficient of the x term, by 2 to get 101. Then add the square of 101 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+202y+101^{2}=-13598+101^{2}$$

Square 101.

$$y^{2}+202y+10201=-13598+10201$$

Add -13598 to 10201.

$$y^{2}+202y+10201=-3397$$

Factor y^{2}+202y+10201. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+101\right)^{2}=-3397$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+101\right)^{2}}=\sqrt{-3397}$$

Simplify.

$$y+101=\sqrt{3397}i y+101=-\sqrt{3397}i$$

Subtract 101 from both sides of the equation.

$$y=-101+\sqrt{3397}i y=-\sqrt{3397}i-101$$