23x^2+6x-3=0

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


Simplifying
23x2 + 6x + -3 = 0

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

Solving
-3 + 6x + 23x2 = 0

Solving for variable 'x'.

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

Divide each side by '23'.
-0.1304347826 + 0.2608695652x + x2 = 0

Move the constant term to the right:

Add '0.1304347826' to each side of the equation.
-0.1304347826 + 0.2608695652x + 0.1304347826 + x2 = 0 + 0.1304347826

Reorder the terms:
-0.1304347826 + 0.1304347826 + 0.2608695652x + x2 = 0 + 0.1304347826

Combine like terms: -0.1304347826 + 0.1304347826 = 0.0000000000
0.0000000000 + 0.2608695652x + x2 = 0 + 0.1304347826
0.2608695652x + x2 = 0 + 0.1304347826

Combine like terms: 0 + 0.1304347826 = 0.1304347826
0.2608695652x + x2 = 0.1304347826

The x term is 0.2608695652x.  Take half its coefficient (0.1304347826).
Square it (0.01701323251) and add it to both sides.

Add '0.01701323251' to each side of the equation.
0.2608695652x + 0.01701323251 + x2 = 0.1304347826 + 0.01701323251

Reorder the terms:
0.01701323251 + 0.2608695652x + x2 = 0.1304347826 + 0.01701323251

Combine like terms: 0.1304347826 + 0.01701323251 = 0.14744801511
0.01701323251 + 0.2608695652x + x2 = 0.14744801511

Factor a perfect square on the left side:
(x + 0.1304347826)(x + 0.1304347826) = 0.14744801511

Calculate the square root of the right side: 0.383989603

Break this problem into two subproblems by setting 
(x + 0.1304347826) equal to 0.383989603 and -0.383989603.

Subproblem 1

x + 0.1304347826 = 0.383989603 Simplifying x + 0.1304347826 = 0.383989603 Reorder the terms: 0.1304347826 + x = 0.383989603 Solving 0.1304347826 + x = 0.383989603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1304347826' to each side of the equation. 0.1304347826 + -0.1304347826 + x = 0.383989603 + -0.1304347826 Combine like terms: 0.1304347826 + -0.1304347826 = 0.0000000000 0.0000000000 + x = 0.383989603 + -0.1304347826 x = 0.383989603 + -0.1304347826 Combine like terms: 0.383989603 + -0.1304347826 = 0.2535548204 x = 0.2535548204 Simplifying x = 0.2535548204

Subproblem 2

x + 0.1304347826 = -0.383989603 Simplifying x + 0.1304347826 = -0.383989603 Reorder the terms: 0.1304347826 + x = -0.383989603 Solving 0.1304347826 + x = -0.383989603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1304347826' to each side of the equation. 0.1304347826 + -0.1304347826 + x = -0.383989603 + -0.1304347826 Combine like terms: 0.1304347826 + -0.1304347826 = 0.0000000000 0.0000000000 + x = -0.383989603 + -0.1304347826 x = -0.383989603 + -0.1304347826 Combine like terms: -0.383989603 + -0.1304347826 = -0.5144243856 x = -0.5144243856 Simplifying x = -0.5144243856

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.2535548204, -0.5144243856}

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