x^2+x=52

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Solution for x^2+x=52 equation:


Simplifying
x2 + x = 52

Reorder the terms:
x + x2 = 52

Solving
x + x2 = 52

Solving for variable 'x'.

Reorder the terms:
-52 + x + x2 = 52 + -52

Combine like terms: 52 + -52 = 0
-52 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '52' to each side of the equation.
-52 + x + 52 + x2 = 0 + 52

Reorder the terms:
-52 + 52 + x + x2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + x + x2 = 0 + 52
x + x2 = 0 + 52

Combine like terms: 0 + 52 = 52
x + x2 = 52

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 = 52 + 0.25

Reorder the terms:
0.25 + x + x2 = 52 + 0.25

Combine like terms: 52 + 0.25 = 52.25
0.25 + x + x2 = 52.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 52.25

Calculate the square root of the right side: 7.228416147

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.228416147 and -7.228416147.

Subproblem 1

x + 0.5 = 7.228416147 Simplifying x + 0.5 = 7.228416147 Reorder the terms: 0.5 + x = 7.228416147 Solving 0.5 + x = 7.228416147 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 = 7.228416147 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.228416147 + -0.5 x = 7.228416147 + -0.5 Combine like terms: 7.228416147 + -0.5 = 6.728416147 x = 6.728416147 Simplifying x = 6.728416147

Subproblem 2

x + 0.5 = -7.228416147 Simplifying x + 0.5 = -7.228416147 Reorder the terms: 0.5 + x = -7.228416147 Solving 0.5 + x = -7.228416147 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 = -7.228416147 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.228416147 + -0.5 x = -7.228416147 + -0.5 Combine like terms: -7.228416147 + -0.5 = -7.728416147 x = -7.728416147 Simplifying x = -7.728416147

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.728416147, -7.728416147}

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