11x^2+23x-2=0

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Solution for 11x^2+23x-2=0 equation:


Simplifying
11x2 + 23x + -2 = 0

Reorder the terms:
-2 + 23x + 11x2 = 0

Solving
-2 + 23x + 11x2 = 0

Solving for variable 'x'.

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

Divide each side by '11'.
-0.1818181818 + 2.090909091x + x2 = 0

Move the constant term to the right:

Add '0.1818181818' to each side of the equation.
-0.1818181818 + 2.090909091x + 0.1818181818 + x2 = 0 + 0.1818181818

Reorder the terms:
-0.1818181818 + 0.1818181818 + 2.090909091x + x2 = 0 + 0.1818181818

Combine like terms: -0.1818181818 + 0.1818181818 = 0.0000000000
0.0000000000 + 2.090909091x + x2 = 0 + 0.1818181818
2.090909091x + x2 = 0 + 0.1818181818

Combine like terms: 0 + 0.1818181818 = 0.1818181818
2.090909091x + x2 = 0.1818181818

The x term is 2.090909091x.  Take half its coefficient (1.045454546).
Square it (1.092975208) and add it to both sides.

Add '1.092975208' to each side of the equation.
2.090909091x + 1.092975208 + x2 = 0.1818181818 + 1.092975208

Reorder the terms:
1.092975208 + 2.090909091x + x2 = 0.1818181818 + 1.092975208

Combine like terms: 0.1818181818 + 1.092975208 = 1.2747933898
1.092975208 + 2.090909091x + x2 = 1.2747933898

Factor a perfect square on the left side:
(x + 1.045454546)(x + 1.045454546) = 1.2747933898

Calculate the square root of the right side: 1.129067487

Break this problem into two subproblems by setting 
(x + 1.045454546) equal to 1.129067487 and -1.129067487.

Subproblem 1

x + 1.045454546 = 1.129067487 Simplifying x + 1.045454546 = 1.129067487 Reorder the terms: 1.045454546 + x = 1.129067487 Solving 1.045454546 + x = 1.129067487 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.045454546' to each side of the equation. 1.045454546 + -1.045454546 + x = 1.129067487 + -1.045454546 Combine like terms: 1.045454546 + -1.045454546 = 0.000000000 0.000000000 + x = 1.129067487 + -1.045454546 x = 1.129067487 + -1.045454546 Combine like terms: 1.129067487 + -1.045454546 = 0.083612941 x = 0.083612941 Simplifying x = 0.083612941

Subproblem 2

x + 1.045454546 = -1.129067487 Simplifying x + 1.045454546 = -1.129067487 Reorder the terms: 1.045454546 + x = -1.129067487 Solving 1.045454546 + x = -1.129067487 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.045454546' to each side of the equation. 1.045454546 + -1.045454546 + x = -1.129067487 + -1.045454546 Combine like terms: 1.045454546 + -1.045454546 = 0.000000000 0.000000000 + x = -1.129067487 + -1.045454546 x = -1.129067487 + -1.045454546 Combine like terms: -1.129067487 + -1.045454546 = -2.174522033 x = -2.174522033 Simplifying x = -2.174522033

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.083612941, -2.174522033}

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