5x-2x-6=-3x^2+6

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Solution for 5x-2x-6=-3x^2+6 equation: 


Simplifying
5x + -2x + -6 = -3x2 + 6

Reorder the terms:
-6 + 5x + -2x = -3x2 + 6

Combine like terms: 5x + -2x = 3x
-6 + 3x = -3x2 + 6

Reorder the terms:
-6 + 3x = 6 + -3x2

Solving
-6 + 3x = 6 + -3x2

Solving for variable 'x'.

Reorder the terms:
-6 + -6 + 3x + 3x2 = 6 + -3x2 + -6 + 3x2

Combine like terms: -6 + -6 = -12
-12 + 3x + 3x2 = 6 + -3x2 + -6 + 3x2

Reorder the terms:
-12 + 3x + 3x2 = 6 + -6 + -3x2 + 3x2

Combine like terms: 6 + -6 = 0
-12 + 3x + 3x2 = 0 + -3x2 + 3x2
-12 + 3x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-12 + 3x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-4 + x + x2)' equal to zero and attempt to solve:

Simplifying
-4 + x + x2 = 0

Solving
-4 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + x + x2 = 0 + 4
x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
x + x2 = 4

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 = 4 + 0.25

Reorder the terms:
0.25 + x + x2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + x + x2 = 4.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
x + 0.5 = 2.061552813

Simplifying
x + 0.5 = 2.061552813

Reorder the terms:
0.5 + x = 2.061552813

Solving
0.5 + x = 2.061552813

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 = 2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.061552813 + -0.5
x = 2.061552813 + -0.5

Combine like terms: 2.061552813 + -0.5 = 1.561552813
x = 1.561552813

Simplifying
x = 1.561552813


Subproblem 2
x + 0.5 = -2.061552813

Simplifying
x + 0.5 = -2.061552813

Reorder the terms:
0.5 + x = -2.061552813

Solving
0.5 + x = -2.061552813

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 = -2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.061552813 + -0.5
x = -2.061552813 + -0.5

Combine like terms: -2.061552813 + -0.5 = -2.561552813
x = -2.561552813

Simplifying
x = -2.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.561552813, -2.561552813}
Solution
x = {1.561552813, -2.561552813}

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