x^2-24=-x

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Solution for x^2-24=-x equation:


Simplifying
x2 + -24 = -1x

Reorder the terms:
-24 + x2 = -1x

Solving
-24 + x2 = -1x

Solving for variable 'x'.

Reorder the terms:
-24 + x + x2 = -1x + x

Combine like terms: -1x + x = 0
-24 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + x + x2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + x + x2 = 0 + 24
x + x2 = 0 + 24

Combine like terms: 0 + 24 = 24
x + x2 = 24

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 = 24 + 0.25

Reorder the terms:
0.25 + x + x2 = 24 + 0.25

Combine like terms: 24 + 0.25 = 24.25
0.25 + x + x2 = 24.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

x + 0.5 = 4.924428901 Simplifying x + 0.5 = 4.924428901 Reorder the terms: 0.5 + x = 4.924428901 Solving 0.5 + x = 4.924428901 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 = 4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.924428901 + -0.5 x = 4.924428901 + -0.5 Combine like terms: 4.924428901 + -0.5 = 4.424428901 x = 4.424428901 Simplifying x = 4.424428901

Subproblem 2

x + 0.5 = -4.924428901 Simplifying x + 0.5 = -4.924428901 Reorder the terms: 0.5 + x = -4.924428901 Solving 0.5 + x = -4.924428901 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 = -4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.924428901 + -0.5 x = -4.924428901 + -0.5 Combine like terms: -4.924428901 + -0.5 = -5.424428901 x = -5.424428901 Simplifying x = -5.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.424428901, -5.424428901}

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