312x-2x^2=2000

Solution for 312x-2x^2=2000 equation:

Simplifying
312x + -2x2 = 2000

Solving
312x + -2x2 = 2000

Solving for variable 'x'.

Reorder the terms:
-2000 + 312x + -2x2 = 2000 + -2000

Combine like terms: 2000 + -2000 = 0
-2000 + 312x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1000 + 156x + -1x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1000 + 156x + -1x2)' equal to zero and attempt to solve:

Simplifying
-1000 + 156x + -1x2 = 0

Solving
-1000 + 156x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1000 + -156x + x2 = 0

Move the constant term to the right:

Add '-1000' to each side of the equation.
1000 + -156x + -1000 + x2 = 0 + -1000

Reorder the terms:
1000 + -1000 + -156x + x2 = 0 + -1000

Combine like terms: 1000 + -1000 = 0
0 + -156x + x2 = 0 + -1000
-156x + x2 = 0 + -1000

Combine like terms: 0 + -1000 = -1000
-156x + x2 = -1000

The x term is -156x.  Take half its coefficient (-78).
Square it (6084) and add it to both sides.

Add '6084' to each side of the equation.
-156x + 6084 + x2 = -1000 + 6084

Reorder the terms:
6084 + -156x + x2 = -1000 + 6084

Combine like terms: -1000 + 6084 = 5084
6084 + -156x + x2 = 5084

Factor a perfect square on the left side:
(x + -78)(x + -78) = 5084

Calculate the square root of the right side: 71.30217388

Break this problem into two subproblems by setting 
(x + -78) equal to 71.30217388 and -71.30217388.

Subproblem 1
x + -78 = 71.30217388

Simplifying
x + -78 = 71.30217388

Reorder the terms:
-78 + x = 71.30217388

Solving
-78 + x = 71.30217388

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '78' to each side of the equation.
-78 + 78 + x = 71.30217388 + 78

Combine like terms: -78 + 78 = 0
0 + x = 71.30217388 + 78
x = 71.30217388 + 78

Combine like terms: 71.30217388 + 78 = 149.30217388
x = 149.30217388

Simplifying
x = 149.30217388

Subproblem 2
x + -78 = -71.30217388

Simplifying
x + -78 = -71.30217388

Reorder the terms:
-78 + x = -71.30217388

Solving
-78 + x = -71.30217388

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '78' to each side of the equation.
-78 + 78 + x = -71.30217388 + 78

Combine like terms: -78 + 78 = 0
0 + x = -71.30217388 + 78
x = -71.30217388 + 78

Combine like terms: -71.30217388 + 78 = 6.69782612
x = 6.69782612

Simplifying
x = 6.69782612

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {149.30217388, 6.69782612}
Solution
x = {149.30217388, 6.69782612}