3x^2-2000x+9=0

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Solution for 3x^2-2000x+9=0 equation:


Simplifying
3x2 + -2000x + 9 = 0

Reorder the terms:
9 + -2000x + 3x2 = 0

Solving
9 + -2000x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
3 + -666.6666667x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -666.6666667x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -666.6666667x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -666.6666667x + x2 = 0 + -3
-666.6666667x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-666.6666667x + x2 = -3

The x term is -666.6666667x.  Take half its coefficient (-333.3333334).
Square it (111111.1112) and add it to both sides.

Add '111111.1112' to each side of the equation.
-666.6666667x + 111111.1112 + x2 = -3 + 111111.1112

Reorder the terms:
111111.1112 + -666.6666667x + x2 = -3 + 111111.1112

Combine like terms: -3 + 111111.1112 = 111108.1112
111111.1112 + -666.6666667x + x2 = 111108.1112

Factor a perfect square on the left side:
(x + -333.3333334)(x + -333.3333334) = 111108.1112

Calculate the square root of the right side: 333.328833436

Break this problem into two subproblems by setting 
(x + -333.3333334) equal to 333.328833436 and -333.328833436.

Subproblem 1

x + -333.3333334 = 333.328833436 Simplifying x + -333.3333334 = 333.328833436 Reorder the terms: -333.3333334 + x = 333.328833436 Solving -333.3333334 + x = 333.328833436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '333.3333334' to each side of the equation. -333.3333334 + 333.3333334 + x = 333.328833436 + 333.3333334 Combine like terms: -333.3333334 + 333.3333334 = 0.0000000 0.0000000 + x = 333.328833436 + 333.3333334 x = 333.328833436 + 333.3333334 Combine like terms: 333.328833436 + 333.3333334 = 666.662166836 x = 666.662166836 Simplifying x = 666.662166836

Subproblem 2

x + -333.3333334 = -333.328833436 Simplifying x + -333.3333334 = -333.328833436 Reorder the terms: -333.3333334 + x = -333.328833436 Solving -333.3333334 + x = -333.328833436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '333.3333334' to each side of the equation. -333.3333334 + 333.3333334 + x = -333.328833436 + 333.3333334 Combine like terms: -333.3333334 + 333.3333334 = 0.0000000 0.0000000 + x = -333.328833436 + 333.3333334 x = -333.328833436 + 333.3333334 Combine like terms: -333.328833436 + 333.3333334 = 0.004499964 x = 0.004499964 Simplifying x = 0.004499964

Solution

The solution to the problem is based on the solutions from the subproblems. x = {666.662166836, 0.004499964}

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