2x^2-12x-23=0

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Solution for 2x^2-12x-23=0 equation:


Simplifying
2x2 + -12x + -23 = 0

Reorder the terms:
-23 + -12x + 2x2 = 0

Solving
-23 + -12x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-11.5 + -6x + x2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + -6x + 11.5 + x2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + -6x + x2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + -6x + x2 = 0 + 11.5
-6x + x2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
-6x + x2 = 11.5

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = 11.5 + 9

Reorder the terms:
9 + -6x + x2 = 11.5 + 9

Combine like terms: 11.5 + 9 = 20.5
9 + -6x + x2 = 20.5

Factor a perfect square on the left side:
(x + -3)(x + -3) = 20.5

Calculate the square root of the right side: 4.527692569

Break this problem into two subproblems by setting 
(x + -3) equal to 4.527692569 and -4.527692569.

Subproblem 1

x + -3 = 4.527692569 Simplifying x + -3 = 4.527692569 Reorder the terms: -3 + x = 4.527692569 Solving -3 + x = 4.527692569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 4.527692569 + 3 Combine like terms: -3 + 3 = 0 0 + x = 4.527692569 + 3 x = 4.527692569 + 3 Combine like terms: 4.527692569 + 3 = 7.527692569 x = 7.527692569 Simplifying x = 7.527692569

Subproblem 2

x + -3 = -4.527692569 Simplifying x + -3 = -4.527692569 Reorder the terms: -3 + x = -4.527692569 Solving -3 + x = -4.527692569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -4.527692569 + 3 Combine like terms: -3 + 3 = 0 0 + x = -4.527692569 + 3 x = -4.527692569 + 3 Combine like terms: -4.527692569 + 3 = -1.527692569 x = -1.527692569 Simplifying x = -1.527692569

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.527692569, -1.527692569}

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