Solve (y+505)^2+y^2+505y+y^2=4

$$( y + 505 ) ^ { 2 } + y ^ { 2 } + 505 y + y ^ { 2 } = 4$$

Solve for y

$$y=\frac{-505+\sqrt{85003}i}{2}\approx -252.5+145.776369827i$$

$$y=\frac{-\sqrt{85003}i-505}{2}\approx -252.5-145.776369827i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+505\right)^{2}.

$$y^{2}+1010y+255025+y^{2}+505y+y^{2}=4$$

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

$$2y^{2}+1010y+255025+505y+y^{2}=4$$

Combine 1010y and 505y to get 1515y.

$$2y^{2}+1515y+255025+y^{2}=4$$

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

$$3y^{2}+1515y+255025=4$$

Subtract 4 from both sides.

$$3y^{2}+1515y+255025-4=0$$

Subtract 4 from 255025 to get 255021.

$$3y^{2}+1515y+255021=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 1515 for b, and 255021 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-1515±\sqrt{1515^{2}-4\times 3\times 255021}}{2\times 3}$$

Square 1515.

$$y=\frac{-1515±\sqrt{2295225-4\times 3\times 255021}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-1515±\sqrt{2295225-12\times 255021}}{2\times 3}$$

Multiply -12 times 255021.

$$y=\frac{-1515±\sqrt{2295225-3060252}}{2\times 3}$$

Add 2295225 to -3060252.

$$y=\frac{-1515±\sqrt{-765027}}{2\times 3}$$

Take the square root of -765027.

$$y=\frac{-1515±3\sqrt{85003}i}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{-1515±3\sqrt{85003}i}{6}$$

Now solve the equation y=\frac{-1515±3\sqrt{85003}i}{6} when ± is plus. Add -1515 to 3i\sqrt{85003}.

$$y=\frac{-1515+3\sqrt{85003}i}{6}$$

Divide -1515+3i\sqrt{85003} by 6.

$$y=\frac{-505+\sqrt{85003}i}{2}$$

Now solve the equation y=\frac{-1515±3\sqrt{85003}i}{6} when ± is minus. Subtract 3i\sqrt{85003} from -1515.

$$y=\frac{-3\sqrt{85003}i-1515}{6}$$

Divide -1515-3i\sqrt{85003} by 6.

$$y=\frac{-\sqrt{85003}i-505}{2}$$

The equation is now solved.

$$y=\frac{-505+\sqrt{85003}i}{2} y=\frac{-\sqrt{85003}i-505}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+505\right)^{2}.

$$y^{2}+1010y+255025+y^{2}+505y+y^{2}=4$$

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

$$2y^{2}+1010y+255025+505y+y^{2}=4$$

Combine 1010y and 505y to get 1515y.

$$2y^{2}+1515y+255025+y^{2}=4$$

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

$$3y^{2}+1515y+255025=4$$

Subtract 255025 from both sides.

$$3y^{2}+1515y=4-255025$$

Subtract 255025 from 4 to get -255021.

$$3y^{2}+1515y=-255021$$

Divide both sides by 3.

$$\frac{3y^{2}+1515y}{3}=-\frac{255021}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}+\frac{1515}{3}y=-\frac{255021}{3}$$

Divide 1515 by 3.

$$y^{2}+505y=-\frac{255021}{3}$$

Divide -255021 by 3.

$$y^{2}+505y=-85007$$

Divide 505, the coefficient of the x term, by 2 to get \frac{505}{2}. Then add the square of \frac{505}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+505y+\left(\frac{505}{2}\right)^{2}=-85007+\left(\frac{505}{2}\right)^{2}$$

Square \frac{505}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+505y+\frac{255025}{4}=-85007+\frac{255025}{4}$$

Add -85007 to \frac{255025}{4}.

$$y^{2}+505y+\frac{255025}{4}=-\frac{85003}{4}$$

Factor y^{2}+505y+\frac{255025}{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}.

$$\left(y+\frac{505}{2}\right)^{2}=-\frac{85003}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{505}{2}\right)^{2}}=\sqrt{-\frac{85003}{4}}$$

Simplify.

$$y+\frac{505}{2}=\frac{\sqrt{85003}i}{2} y+\frac{505}{2}=-\frac{\sqrt{85003}i}{2}$$

Subtract \frac{505}{2} from both sides of the equation.

$$y=\frac{-505+\sqrt{85003}i}{2} y=\frac{-\sqrt{85003}i-505}{2}$$