2x^2-x+3=3x+7

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Solution for 2x^2-x+3=3x+7 equation: 


Simplifying
2x2 + -1x + 3 = 3x + 7

Reorder the terms:
3 + -1x + 2x2 = 3x + 7

Reorder the terms:
3 + -1x + 2x2 = 7 + 3x

Solving
3 + -1x + 2x2 = 7 + 3x

Solving for variable 'x'.

Reorder the terms:
3 + -7 + -1x + -3x + 2x2 = 7 + 3x + -7 + -3x

Combine like terms: 3 + -7 = -4
-4 + -1x + -3x + 2x2 = 7 + 3x + -7 + -3x

Combine like terms: -1x + -3x = -4x
-4 + -4x + 2x2 = 7 + 3x + -7 + -3x

Reorder the terms:
-4 + -4x + 2x2 = 7 + -7 + 3x + -3x

Combine like terms: 7 + -7 = 0
-4 + -4x + 2x2 = 0 + 3x + -3x
-4 + -4x + 2x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-4 + -4x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + -2x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2 + -2x + x2)' equal to zero and attempt to solve:

Simplifying
-2 + -2x + x2 = 0

Solving
-2 + -2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -2x + x2 = 0 + 2

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

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

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 2 + 1

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

Combine like terms: 2 + 1 = 3
1 + -2x + x2 = 3

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -1) equal to 1.732050808 and -1.732050808.


Subproblem 1
x + -1 = 1.732050808

Simplifying
x + -1 = 1.732050808

Reorder the terms:
-1 + x = 1.732050808

Solving
-1 + x = 1.732050808

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.732050808 + 1
x = 1.732050808 + 1

Combine like terms: 1.732050808 + 1 = 2.732050808
x = 2.732050808

Simplifying
x = 2.732050808


Subproblem 2
x + -1 = -1.732050808

Simplifying
x + -1 = -1.732050808

Reorder the terms:
-1 + x = -1.732050808

Solving
-1 + x = -1.732050808

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.732050808 + 1
x = -1.732050808 + 1

Combine like terms: -1.732050808 + 1 = -0.732050808
x = -0.732050808

Simplifying
x = -0.732050808


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.732050808, -0.732050808}
Solution
x = {2.732050808, -0.732050808}

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