x^2+x=624

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Solution for x^2+x=624 equation:


Simplifying
x2 + x = 624

Reorder the terms:
x + x2 = 624

Solving
x + x2 = 624

Solving for variable 'x'.

Reorder the terms:
-624 + x + x2 = 624 + -624

Combine like terms: 624 + -624 = 0
-624 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '624' to each side of the equation.
-624 + x + 624 + x2 = 0 + 624

Reorder the terms:
-624 + 624 + x + x2 = 0 + 624

Combine like terms: -624 + 624 = 0
0 + x + x2 = 0 + 624
x + x2 = 0 + 624

Combine like terms: 0 + 624 = 624
x + x2 = 624

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 = 624 + 0.25

Reorder the terms:
0.25 + x + x2 = 624 + 0.25

Combine like terms: 624 + 0.25 = 624.25
0.25 + x + x2 = 624.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 624.25

Calculate the square root of the right side: 24.984995497

Break this problem into two subproblems by setting 
(x + 0.5) equal to 24.984995497 and -24.984995497.

Subproblem 1

x + 0.5 = 24.984995497 Simplifying x + 0.5 = 24.984995497 Reorder the terms: 0.5 + x = 24.984995497 Solving 0.5 + x = 24.984995497 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 = 24.984995497 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 24.984995497 + -0.5 x = 24.984995497 + -0.5 Combine like terms: 24.984995497 + -0.5 = 24.484995497 x = 24.484995497 Simplifying x = 24.484995497

Subproblem 2

x + 0.5 = -24.984995497 Simplifying x + 0.5 = -24.984995497 Reorder the terms: 0.5 + x = -24.984995497 Solving 0.5 + x = -24.984995497 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 = -24.984995497 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -24.984995497 + -0.5 x = -24.984995497 + -0.5 Combine like terms: -24.984995497 + -0.5 = -25.484995497 x = -25.484995497 Simplifying x = -25.484995497

Solution

The solution to the problem is based on the solutions from the subproblems. x = {24.484995497, -25.484995497}

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