51x^2-500x=1000

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Solution for 51x^2-500x=1000 equation:


Simplifying
51x2 + -500x = 1000

Reorder the terms:
-500x + 51x2 = 1000

Solving
-500x + 51x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + -500x + 51x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + -500x + 51x2 = 0

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

Divide each side by '51'.
-19.60784314 + -9.803921569x + x2 = 0

Move the constant term to the right:

Add '19.60784314' to each side of the equation.
-19.60784314 + -9.803921569x + 19.60784314 + x2 = 0 + 19.60784314

Reorder the terms:
-19.60784314 + 19.60784314 + -9.803921569x + x2 = 0 + 19.60784314

Combine like terms: -19.60784314 + 19.60784314 = 0.00000000
0.00000000 + -9.803921569x + x2 = 0 + 19.60784314
-9.803921569x + x2 = 0 + 19.60784314

Combine like terms: 0 + 19.60784314 = 19.60784314
-9.803921569x + x2 = 19.60784314

The x term is -9.803921569x.  Take half its coefficient (-4.901960785).
Square it (24.02921954) and add it to both sides.

Add '24.02921954' to each side of the equation.
-9.803921569x + 24.02921954 + x2 = 19.60784314 + 24.02921954

Reorder the terms:
24.02921954 + -9.803921569x + x2 = 19.60784314 + 24.02921954

Combine like terms: 19.60784314 + 24.02921954 = 43.63706268
24.02921954 + -9.803921569x + x2 = 43.63706268

Factor a perfect square on the left side:
(x + -4.901960785)(x + -4.901960785) = 43.63706268

Calculate the square root of the right side: 6.605835502

Break this problem into two subproblems by setting 
(x + -4.901960785) equal to 6.605835502 and -6.605835502.

Subproblem 1

x + -4.901960785 = 6.605835502 Simplifying x + -4.901960785 = 6.605835502 Reorder the terms: -4.901960785 + x = 6.605835502 Solving -4.901960785 + x = 6.605835502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.901960785' to each side of the equation. -4.901960785 + 4.901960785 + x = 6.605835502 + 4.901960785 Combine like terms: -4.901960785 + 4.901960785 = 0.000000000 0.000000000 + x = 6.605835502 + 4.901960785 x = 6.605835502 + 4.901960785 Combine like terms: 6.605835502 + 4.901960785 = 11.507796287 x = 11.507796287 Simplifying x = 11.507796287

Subproblem 2

x + -4.901960785 = -6.605835502 Simplifying x + -4.901960785 = -6.605835502 Reorder the terms: -4.901960785 + x = -6.605835502 Solving -4.901960785 + x = -6.605835502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.901960785' to each side of the equation. -4.901960785 + 4.901960785 + x = -6.605835502 + 4.901960785 Combine like terms: -4.901960785 + 4.901960785 = 0.000000000 0.000000000 + x = -6.605835502 + 4.901960785 x = -6.605835502 + 4.901960785 Combine like terms: -6.605835502 + 4.901960785 = -1.703874717 x = -1.703874717 Simplifying x = -1.703874717

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.507796287, -1.703874717}

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