x+x^2=24+2x

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


Simplifying
x + x2 = 24 + 2x

Solving
x + x2 = 24 + 2x

Solving for variable 'x'.

Reorder the terms:
-24 + x + -2x + x2 = 24 + 2x + -24 + -2x

Combine like terms: x + -2x = -1x
-24 + -1x + x2 = 24 + 2x + -24 + -2x

Reorder the terms:
-24 + -1x + x2 = 24 + -24 + 2x + -2x

Combine like terms: 24 + -24 = 0
-24 + -1x + x2 = 0 + 2x + -2x
-24 + -1x + x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-24 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -1x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + -1x + x2 = 0 + 24

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

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

The x term is -1x.  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.
-1x + 0.25 + x2 = 24 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 24 + 0.25

Combine like terms: 24 + 0.25 = 24.25
0.25 + -1x + 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 = 5.424428901 x = 5.424428901 Simplifying x = 5.424428901

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 = -4.424428901 x = -4.424428901 Simplifying x = -4.424428901

Solution

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

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