x^2-x-61=0

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Solution for x^2-x-61=0 equation:


Simplifying
x2 + -1x + -61 = 0

Reorder the terms:
-61 + -1x + x2 = 0

Solving
-61 + -1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '61' to each side of the equation.
-61 + -1x + 61 + x2 = 0 + 61

Reorder the terms:
-61 + 61 + -1x + x2 = 0 + 61

Combine like terms: -61 + 61 = 0
0 + -1x + x2 = 0 + 61
-1x + x2 = 0 + 61

Combine like terms: 0 + 61 = 61
-1x + x2 = 61

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 = 61 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 61 + 0.25

Combine like terms: 61 + 0.25 = 61.25
0.25 + -1x + x2 = 61.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 61.25

Calculate the square root of the right side: 7.826237921

Break this problem into two subproblems by setting 
(x + -0.5) equal to 7.826237921 and -7.826237921.

Subproblem 1

x + -0.5 = 7.826237921 Simplifying x + -0.5 = 7.826237921 Reorder the terms: -0.5 + x = 7.826237921 Solving -0.5 + x = 7.826237921 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 = 7.826237921 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 7.826237921 + 0.5 x = 7.826237921 + 0.5 Combine like terms: 7.826237921 + 0.5 = 8.326237921 x = 8.326237921 Simplifying x = 8.326237921

Subproblem 2

x + -0.5 = -7.826237921 Simplifying x + -0.5 = -7.826237921 Reorder the terms: -0.5 + x = -7.826237921 Solving -0.5 + x = -7.826237921 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 = -7.826237921 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -7.826237921 + 0.5 x = -7.826237921 + 0.5 Combine like terms: -7.826237921 + 0.5 = -7.326237921 x = -7.326237921 Simplifying x = -7.326237921

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.326237921, -7.326237921}

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