6x^2-23x-3=0

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Solution for 6x^2-23x-3=0 equation:


Simplifying
6x2 + -23x + -3 = 0

Reorder the terms:
-3 + -23x + 6x2 = 0

Solving
-3 + -23x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-0.5 + -3.833333333x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -3.833333333x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -3.833333333x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -3.833333333x + x2 = 0 + 0.5
-3.833333333x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-3.833333333x + x2 = 0.5

The x term is -3.833333333x.  Take half its coefficient (-1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
-3.833333333x + 3.673611112 + x2 = 0.5 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333x + x2 = 0.5 + 3.673611112

Combine like terms: 0.5 + 3.673611112 = 4.173611112
3.673611112 + -3.833333333x + x2 = 4.173611112

Factor a perfect square on the left side:
(x + -1.916666667)(x + -1.916666667) = 4.173611112

Calculate the square root of the right side: 2.042941779

Break this problem into two subproblems by setting 
(x + -1.916666667) equal to 2.042941779 and -2.042941779.

Subproblem 1

x + -1.916666667 = 2.042941779 Simplifying x + -1.916666667 = 2.042941779 Reorder the terms: -1.916666667 + x = 2.042941779 Solving -1.916666667 + x = 2.042941779 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + x = 2.042941779 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + x = 2.042941779 + 1.916666667 x = 2.042941779 + 1.916666667 Combine like terms: 2.042941779 + 1.916666667 = 3.959608446 x = 3.959608446 Simplifying x = 3.959608446

Subproblem 2

x + -1.916666667 = -2.042941779 Simplifying x + -1.916666667 = -2.042941779 Reorder the terms: -1.916666667 + x = -2.042941779 Solving -1.916666667 + x = -2.042941779 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + x = -2.042941779 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + x = -2.042941779 + 1.916666667 x = -2.042941779 + 1.916666667 Combine like terms: -2.042941779 + 1.916666667 = -0.126275112 x = -0.126275112 Simplifying x = -0.126275112

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.959608446, -0.126275112}

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