123=2x*x+4x-31

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Solution for 123=2x*x+4x-31 equation: 


Simplifying
123 = 2x * x + 4x + -31

Multiply x * x
123 = 2x2 + 4x + -31

Reorder the terms:
123 = -31 + 4x + 2x2

Solving
123 = -31 + 4x + 2x2

Solving for variable 'x'.

Combine like terms: 123 + 31 = 154
154 + -4x + -2x2 = -31 + 4x + 2x2 + 31 + -4x + -2x2

Reorder the terms:
154 + -4x + -2x2 = -31 + 31 + 4x + -4x + 2x2 + -2x2

Combine like terms: -31 + 31 = 0
154 + -4x + -2x2 = 0 + 4x + -4x + 2x2 + -2x2
154 + -4x + -2x2 = 4x + -4x + 2x2 + -2x2

Combine like terms: 4x + -4x = 0
154 + -4x + -2x2 = 0 + 2x2 + -2x2
154 + -4x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
154 + -4x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(77 + -2x + -1x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(77 + -2x + -1x2)' equal to zero and attempt to solve:

Simplifying
77 + -2x + -1x2 = 0

Solving
77 + -2x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-77 + 2x + x2 = 0

Move the constant term to the right:

Add '77' to each side of the equation.
-77 + 2x + 77 + x2 = 0 + 77

Reorder the terms:
-77 + 77 + 2x + x2 = 0 + 77

Combine like terms: -77 + 77 = 0
0 + 2x + x2 = 0 + 77
2x + x2 = 0 + 77

Combine like terms: 0 + 77 = 77
2x + x2 = 77

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 = 77 + 1

Reorder the terms:
1 + 2x + x2 = 77 + 1

Combine like terms: 77 + 1 = 78
1 + 2x + x2 = 78

Factor a perfect square on the left side:
(x + 1)(x + 1) = 78

Calculate the square root of the right side: 8.831760866

Break this problem into two subproblems by setting 
(x + 1) equal to 8.831760866 and -8.831760866.


Subproblem 1
x + 1 = 8.831760866

Simplifying
x + 1 = 8.831760866

Reorder the terms:
1 + x = 8.831760866

Solving
1 + x = 8.831760866

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 = 8.831760866 + -1

Combine like terms: 1 + -1 = 0
0 + x = 8.831760866 + -1
x = 8.831760866 + -1

Combine like terms: 8.831760866 + -1 = 7.831760866
x = 7.831760866

Simplifying
x = 7.831760866


Subproblem 2
x + 1 = -8.831760866

Simplifying
x + 1 = -8.831760866

Reorder the terms:
1 + x = -8.831760866

Solving
1 + x = -8.831760866

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 = -8.831760866 + -1

Combine like terms: 1 + -1 = 0
0 + x = -8.831760866 + -1
x = -8.831760866 + -1

Combine like terms: -8.831760866 + -1 = -9.831760866
x = -9.831760866

Simplifying
x = -9.831760866


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.831760866, -9.831760866}
Solution
x = {7.831760866, -9.831760866}

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