x^2+x=53

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Solution for x^2+x=53 equation:


Simplifying
x2 + x = 53

Reorder the terms:
x + x2 = 53

Solving
x + x2 = 53

Solving for variable 'x'.

Reorder the terms:
-53 + x + x2 = 53 + -53

Combine like terms: 53 + -53 = 0
-53 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + x + 53 + x2 = 0 + 53

Reorder the terms:
-53 + 53 + x + x2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + x + x2 = 0 + 53
x + x2 = 0 + 53

Combine like terms: 0 + 53 = 53
x + x2 = 53

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 = 53 + 0.25

Reorder the terms:
0.25 + x + x2 = 53 + 0.25

Combine like terms: 53 + 0.25 = 53.25
0.25 + x + x2 = 53.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 53.25

Calculate the square root of the right side: 7.29725976

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.29725976 and -7.29725976.

Subproblem 1

x + 0.5 = 7.29725976 Simplifying x + 0.5 = 7.29725976 Reorder the terms: 0.5 + x = 7.29725976 Solving 0.5 + x = 7.29725976 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 = 7.29725976 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.29725976 + -0.5 x = 7.29725976 + -0.5 Combine like terms: 7.29725976 + -0.5 = 6.79725976 x = 6.79725976 Simplifying x = 6.79725976

Subproblem 2

x + 0.5 = -7.29725976 Simplifying x + 0.5 = -7.29725976 Reorder the terms: 0.5 + x = -7.29725976 Solving 0.5 + x = -7.29725976 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 = -7.29725976 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.29725976 + -0.5 x = -7.29725976 + -0.5 Combine like terms: -7.29725976 + -0.5 = -7.79725976 x = -7.79725976 Simplifying x = -7.79725976

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.79725976, -7.79725976}

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