-23x^2+x+3=0

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


Simplifying
-23x2 + x + 3 = 0

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

Solving
3 + x + -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 + -0.04347826087x + x2 = 0

Move the constant term to the right:

Add '0.1304347826' to each side of the equation.
-0.1304347826 + -0.04347826087x + 0.1304347826 + x2 = 0 + 0.1304347826

Reorder the terms:
-0.1304347826 + 0.1304347826 + -0.04347826087x + x2 = 0 + 0.1304347826

Combine like terms: -0.1304347826 + 0.1304347826 = 0.0000000000
0.0000000000 + -0.04347826087x + x2 = 0 + 0.1304347826
-0.04347826087x + x2 = 0 + 0.1304347826

Combine like terms: 0 + 0.1304347826 = 0.1304347826
-0.04347826087x + x2 = 0.1304347826

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.04347826087x + 0.25 + x2 = 0.1304347826 + 0.25

Reorder the terms:
0.25 + -0.04347826087x + x2 = 0.1304347826 + 0.25

Combine like terms: 0.1304347826 + 0.25 = 0.3804347826
0.25 + -0.04347826087x + x2 = 0.3804347826

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.3804347826

Calculate the square root of the right side: 0.616793955

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.616793955 and -0.616793955.

Subproblem 1

x + 0.5 = 0.616793955 Simplifying x + 0.5 = 0.616793955 Reorder the terms: 0.5 + x = 0.616793955 Solving 0.5 + x = 0.616793955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.616793955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.616793955 + -0.5 x = 0.616793955 + -0.5 Combine like terms: 0.616793955 + -0.5 = 0.116793955 x = 0.116793955 Simplifying x = 0.116793955

Subproblem 2

x + 0.5 = -0.616793955 Simplifying x + 0.5 = -0.616793955 Reorder the terms: 0.5 + x = -0.616793955 Solving 0.5 + x = -0.616793955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.616793955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.616793955 + -0.5 x = -0.616793955 + -0.5 Combine like terms: -0.616793955 + -0.5 = -1.116793955 x = -1.116793955 Simplifying x = -1.116793955

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.116793955, -1.116793955}

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