x^2+x=51

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Solution for x^2+x=51 equation:


Simplifying
x2 + x = 51

Reorder the terms:
x + x2 = 51

Solving
x + x2 = 51

Solving for variable 'x'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + x + 51 + x2 = 0 + 51

Reorder the terms:
-51 + 51 + x + x2 = 0 + 51

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

Combine like terms: 0 + 51 = 51
x + x2 = 51

The x term is x.  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.
x + 0.25 + x2 = 51 + 0.25

Reorder the terms:
0.25 + x + x2 = 51 + 0.25

Combine like terms: 51 + 0.25 = 51.25
0.25 + x + x2 = 51.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

x + 0.5 = 7.158910532 Simplifying x + 0.5 = 7.158910532 Reorder the terms: 0.5 + x = 7.158910532 Solving 0.5 + x = 7.158910532 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 = 7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.158910532 + -0.5 x = 7.158910532 + -0.5 Combine like terms: 7.158910532 + -0.5 = 6.658910532 x = 6.658910532 Simplifying x = 6.658910532

Subproblem 2

x + 0.5 = -7.158910532 Simplifying x + 0.5 = -7.158910532 Reorder the terms: 0.5 + x = -7.158910532 Solving 0.5 + x = -7.158910532 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 = -7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.158910532 + -0.5 x = -7.158910532 + -0.5 Combine like terms: -7.158910532 + -0.5 = -7.658910532 x = -7.658910532 Simplifying x = -7.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.658910532, -7.658910532}

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