x(x-1)=513

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Solution for x(x-1)=513 equation:


Simplifying
x(x + -1) = 513

Reorder the terms:
x(-1 + x) = 513
(-1 * x + x * x) = 513
(-1x + x2) = 513

Solving
-1x + x2 = 513

Solving for variable 'x'.

Reorder the terms:
-513 + -1x + x2 = 513 + -513

Combine like terms: 513 + -513 = 0
-513 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '513' to each side of the equation.
-513 + -1x + 513 + x2 = 0 + 513

Reorder the terms:
-513 + 513 + -1x + x2 = 0 + 513

Combine like terms: -513 + 513 = 0
0 + -1x + x2 = 0 + 513
-1x + x2 = 0 + 513

Combine like terms: 0 + 513 = 513
-1x + x2 = 513

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 513 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 513 + 0.25

Combine like terms: 513 + 0.25 = 513.25
0.25 + -1x + x2 = 513.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 513.25

Calculate the square root of the right side: 22.655021518

Break this problem into two subproblems by setting 
(x + -0.5) equal to 22.655021518 and -22.655021518.

Subproblem 1

x + -0.5 = 22.655021518 Simplifying x + -0.5 = 22.655021518 Reorder the terms: -0.5 + x = 22.655021518 Solving -0.5 + x = 22.655021518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 22.655021518 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 22.655021518 + 0.5 x = 22.655021518 + 0.5 Combine like terms: 22.655021518 + 0.5 = 23.155021518 x = 23.155021518 Simplifying x = 23.155021518

Subproblem 2

x + -0.5 = -22.655021518 Simplifying x + -0.5 = -22.655021518 Reorder the terms: -0.5 + x = -22.655021518 Solving -0.5 + x = -22.655021518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -22.655021518 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -22.655021518 + 0.5 x = -22.655021518 + 0.5 Combine like terms: -22.655021518 + 0.5 = -22.155021518 x = -22.155021518 Simplifying x = -22.155021518

Solution

The solution to the problem is based on the solutions from the subproblems. x = {23.155021518, -22.155021518}

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