x=22x+3x*x+24

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Solution for x=22x+3x*x+24 equation: 


Simplifying
x = 22x + 3x * x + 24

Multiply x * x
x = 22x + 3x2 + 24

Reorder the terms:
x = 24 + 22x + 3x2

Solving
x = 24 + 22x + 3x2

Solving for variable 'x'.

Reorder the terms:
-24 + x + -22x + -3x2 = 24 + 22x + 3x2 + -24 + -22x + -3x2

Combine like terms: x + -22x = -21x
-24 + -21x + -3x2 = 24 + 22x + 3x2 + -24 + -22x + -3x2

Reorder the terms:
-24 + -21x + -3x2 = 24 + -24 + 22x + -22x + 3x2 + -3x2

Combine like terms: 24 + -24 = 0
-24 + -21x + -3x2 = 0 + 22x + -22x + 3x2 + -3x2
-24 + -21x + -3x2 = 22x + -22x + 3x2 + -3x2

Combine like terms: 22x + -22x = 0
-24 + -21x + -3x2 = 0 + 3x2 + -3x2
-24 + -21x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-24 + -21x + -3x2 = 0

Factor out the Greatest Common Factor (GCF), '-3'.
-3(8 + 7x + x2) = 0

Ignore the factor -3.

Subproblem 1
Set the factor '(8 + 7x + x2)' equal to zero and attempt to solve:

Simplifying
8 + 7x + x2 = 0

Solving
8 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 7x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 7x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 7x + x2 = 0 + -8
7x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
7x + x2 = -8

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -8 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -8 + 12.25

Combine like terms: -8 + 12.25 = 4.25
12.25 + 7x + x2 = 4.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + 3.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
x + 3.5 = 2.061552813

Simplifying
x + 3.5 = 2.061552813

Reorder the terms:
3.5 + x = 2.061552813

Solving
3.5 + x = 2.061552813

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 2.061552813 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 2.061552813 + -3.5
x = 2.061552813 + -3.5

Combine like terms: 2.061552813 + -3.5 = -1.438447187
x = -1.438447187

Simplifying
x = -1.438447187


Subproblem 2
x + 3.5 = -2.061552813

Simplifying
x + 3.5 = -2.061552813

Reorder the terms:
3.5 + x = -2.061552813

Solving
3.5 + x = -2.061552813

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -2.061552813 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -2.061552813 + -3.5
x = -2.061552813 + -3.5

Combine like terms: -2.061552813 + -3.5 = -5.561552813
x = -5.561552813

Simplifying
x = -5.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.438447187, -5.561552813}
Solution
x = {-1.438447187, -5.561552813}

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