x^2+521x+223=0

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Solution for x^2+521x+223=0 equation:


Simplifying
x2 + 521x + 223 = 0

Reorder the terms:
223 + 521x + x2 = 0

Solving
223 + 521x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-223' to each side of the equation.
223 + 521x + -223 + x2 = 0 + -223

Reorder the terms:
223 + -223 + 521x + x2 = 0 + -223

Combine like terms: 223 + -223 = 0
0 + 521x + x2 = 0 + -223
521x + x2 = 0 + -223

Combine like terms: 0 + -223 = -223
521x + x2 = -223

The x term is 521x.  Take half its coefficient (260.5).
Square it (67860.25) and add it to both sides.

Add '67860.25' to each side of the equation.
521x + 67860.25 + x2 = -223 + 67860.25

Reorder the terms:
67860.25 + 521x + x2 = -223 + 67860.25

Combine like terms: -223 + 67860.25 = 67637.25
67860.25 + 521x + x2 = 67637.25

Factor a perfect square on the left side:
(x + 260.5)(x + 260.5) = 67637.25

Calculate the square root of the right side: 260.07162475

Break this problem into two subproblems by setting 
(x + 260.5) equal to 260.07162475 and -260.07162475.

Subproblem 1

x + 260.5 = 260.07162475 Simplifying x + 260.5 = 260.07162475 Reorder the terms: 260.5 + x = 260.07162475 Solving 260.5 + x = 260.07162475 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-260.5' to each side of the equation. 260.5 + -260.5 + x = 260.07162475 + -260.5 Combine like terms: 260.5 + -260.5 = 0.0 0.0 + x = 260.07162475 + -260.5 x = 260.07162475 + -260.5 Combine like terms: 260.07162475 + -260.5 = -0.42837525 x = -0.42837525 Simplifying x = -0.42837525

Subproblem 2

x + 260.5 = -260.07162475 Simplifying x + 260.5 = -260.07162475 Reorder the terms: 260.5 + x = -260.07162475 Solving 260.5 + x = -260.07162475 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-260.5' to each side of the equation. 260.5 + -260.5 + x = -260.07162475 + -260.5 Combine like terms: 260.5 + -260.5 = 0.0 0.0 + x = -260.07162475 + -260.5 x = -260.07162475 + -260.5 Combine like terms: -260.07162475 + -260.5 = -520.57162475 x = -520.57162475 Simplifying x = -520.57162475

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.42837525, -520.57162475}

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