x^2+x=76

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Solution for x^2+x=76 equation:


Simplifying
x2 + x = 76

Reorder the terms:
x + x2 = 76

Solving
x + x2 = 76

Solving for variable 'x'.

Reorder the terms:
-76 + x + x2 = 76 + -76

Combine like terms: 76 + -76 = 0
-76 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '76' to each side of the equation.
-76 + x + 76 + x2 = 0 + 76

Reorder the terms:
-76 + 76 + x + x2 = 0 + 76

Combine like terms: -76 + 76 = 0
0 + x + x2 = 0 + 76
x + x2 = 0 + 76

Combine like terms: 0 + 76 = 76
x + x2 = 76

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 = 76 + 0.25

Reorder the terms:
0.25 + x + x2 = 76 + 0.25

Combine like terms: 76 + 0.25 = 76.25
0.25 + x + x2 = 76.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 76.25

Calculate the square root of the right side: 8.732124598

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.732124598 and -8.732124598.

Subproblem 1

x + 0.5 = 8.732124598 Simplifying x + 0.5 = 8.732124598 Reorder the terms: 0.5 + x = 8.732124598 Solving 0.5 + x = 8.732124598 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 = 8.732124598 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.732124598 + -0.5 x = 8.732124598 + -0.5 Combine like terms: 8.732124598 + -0.5 = 8.232124598 x = 8.232124598 Simplifying x = 8.232124598

Subproblem 2

x + 0.5 = -8.732124598 Simplifying x + 0.5 = -8.732124598 Reorder the terms: 0.5 + x = -8.732124598 Solving 0.5 + x = -8.732124598 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 = -8.732124598 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.732124598 + -0.5 x = -8.732124598 + -0.5 Combine like terms: -8.732124598 + -0.5 = -9.232124598 x = -9.232124598 Simplifying x = -9.232124598

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.232124598, -9.232124598}

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