x^2+23x+53=0

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Solution for x^2+23x+53=0 equation:


Simplifying
x2 + 23x + 53 = 0

Reorder the terms:
53 + 23x + x2 = 0

Solving
53 + 23x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-53' to each side of the equation.
53 + 23x + -53 + x2 = 0 + -53

Reorder the terms:
53 + -53 + 23x + x2 = 0 + -53

Combine like terms: 53 + -53 = 0
0 + 23x + x2 = 0 + -53
23x + x2 = 0 + -53

Combine like terms: 0 + -53 = -53
23x + x2 = -53

The x term is 23x.  Take half its coefficient (11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
23x + 132.25 + x2 = -53 + 132.25

Reorder the terms:
132.25 + 23x + x2 = -53 + 132.25

Combine like terms: -53 + 132.25 = 79.25
132.25 + 23x + x2 = 79.25

Factor a perfect square on the left side:
(x + 11.5)(x + 11.5) = 79.25

Calculate the square root of the right side: 8.902246907

Break this problem into two subproblems by setting 
(x + 11.5) equal to 8.902246907 and -8.902246907.

Subproblem 1

x + 11.5 = 8.902246907 Simplifying x + 11.5 = 8.902246907 Reorder the terms: 11.5 + x = 8.902246907 Solving 11.5 + x = 8.902246907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = 8.902246907 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = 8.902246907 + -11.5 x = 8.902246907 + -11.5 Combine like terms: 8.902246907 + -11.5 = -2.597753093 x = -2.597753093 Simplifying x = -2.597753093

Subproblem 2

x + 11.5 = -8.902246907 Simplifying x + 11.5 = -8.902246907 Reorder the terms: 11.5 + x = -8.902246907 Solving 11.5 + x = -8.902246907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = -8.902246907 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = -8.902246907 + -11.5 x = -8.902246907 + -11.5 Combine like terms: -8.902246907 + -11.5 = -20.402246907 x = -20.402246907 Simplifying x = -20.402246907

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.597753093, -20.402246907}

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