x^2-x-414=0

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Solution for x^2-x-414=0 equation:


Simplifying
x2 + -1x + -414 = 0

Reorder the terms:
-414 + -1x + x2 = 0

Solving
-414 + -1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '414' to each side of the equation.
-414 + -1x + 414 + x2 = 0 + 414

Reorder the terms:
-414 + 414 + -1x + x2 = 0 + 414

Combine like terms: -414 + 414 = 0
0 + -1x + x2 = 0 + 414
-1x + x2 = 0 + 414

Combine like terms: 0 + 414 = 414
-1x + x2 = 414

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 = 414 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 414 + 0.25

Combine like terms: 414 + 0.25 = 414.25
0.25 + -1x + x2 = 414.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 414.25

Calculate the square root of the right side: 20.353132437

Break this problem into two subproblems by setting 
(x + -0.5) equal to 20.353132437 and -20.353132437.

Subproblem 1

x + -0.5 = 20.353132437 Simplifying x + -0.5 = 20.353132437 Reorder the terms: -0.5 + x = 20.353132437 Solving -0.5 + x = 20.353132437 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 = 20.353132437 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 20.353132437 + 0.5 x = 20.353132437 + 0.5 Combine like terms: 20.353132437 + 0.5 = 20.853132437 x = 20.853132437 Simplifying x = 20.853132437

Subproblem 2

x + -0.5 = -20.353132437 Simplifying x + -0.5 = -20.353132437 Reorder the terms: -0.5 + x = -20.353132437 Solving -0.5 + x = -20.353132437 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 = -20.353132437 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -20.353132437 + 0.5 x = -20.353132437 + 0.5 Combine like terms: -20.353132437 + 0.5 = -19.853132437 x = -19.853132437 Simplifying x = -19.853132437

Solution

The solution to the problem is based on the solutions from the subproblems. x = {20.853132437, -19.853132437}

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