3=1000y-2y^2

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Solution for 3=1000y-2y^2 equation:


Simplifying
3 = 1000y + -2y2

Solving
3 = 1000y + -2y2

Solving for variable 'y'.

Reorder the terms:
3 + -1000y + 2y2 = 1000y + -1000y + -2y2 + 2y2

Combine like terms: 1000y + -1000y = 0
3 + -1000y + 2y2 = 0 + -2y2 + 2y2
3 + -1000y + 2y2 = -2y2 + 2y2

Combine like terms: -2y2 + 2y2 = 0
3 + -1000y + 2y2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
1.5 + -500y + y2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -500y + -1.5 + y2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -500y + y2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -500y + y2 = 0 + -1.5
-500y + y2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-500y + y2 = -1.5

The y term is -500y.  Take half its coefficient (-250).
Square it (62500) and add it to both sides.

Add '62500' to each side of the equation.
-500y + 62500 + y2 = -1.5 + 62500

Reorder the terms:
62500 + -500y + y2 = -1.5 + 62500

Combine like terms: -1.5 + 62500 = 62498.5
62500 + -500y + y2 = 62498.5

Factor a perfect square on the left side:
(y + -250)(y + -250) = 62498.5

Calculate the square root of the right side: 249.996999982

Break this problem into two subproblems by setting 
(y + -250) equal to 249.996999982 and -249.996999982.

Subproblem 1

y + -250 = 249.996999982 Simplifying y + -250 = 249.996999982 Reorder the terms: -250 + y = 249.996999982 Solving -250 + y = 249.996999982 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + y = 249.996999982 + 250 Combine like terms: -250 + 250 = 0 0 + y = 249.996999982 + 250 y = 249.996999982 + 250 Combine like terms: 249.996999982 + 250 = 499.996999982 y = 499.996999982 Simplifying y = 499.996999982

Subproblem 2

y + -250 = -249.996999982 Simplifying y + -250 = -249.996999982 Reorder the terms: -250 + y = -249.996999982 Solving -250 + y = -249.996999982 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + y = -249.996999982 + 250 Combine like terms: -250 + 250 = 0 0 + y = -249.996999982 + 250 y = -249.996999982 + 250 Combine like terms: -249.996999982 + 250 = 0.003000018 y = 0.003000018 Simplifying y = 0.003000018

Solution

The solution to the problem is based on the solutions from the subproblems. y = {499.996999982, 0.003000018}

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