11x^2-23x-21=0

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Solution for 11x^2-23x-21=0 equation:


Simplifying
11x2 + -23x + -21 = 0

Reorder the terms:
-21 + -23x + 11x2 = 0

Solving
-21 + -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'.
-1.909090909 + -2.090909091x + x2 = 0

Move the constant term to the right:

Add '1.909090909' to each side of the equation.
-1.909090909 + -2.090909091x + 1.909090909 + x2 = 0 + 1.909090909

Reorder the terms:
-1.909090909 + 1.909090909 + -2.090909091x + x2 = 0 + 1.909090909

Combine like terms: -1.909090909 + 1.909090909 = 0.000000000
0.000000000 + -2.090909091x + x2 = 0 + 1.909090909
-2.090909091x + x2 = 0 + 1.909090909

Combine like terms: 0 + 1.909090909 = 1.909090909
-2.090909091x + x2 = 1.909090909

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 = 1.909090909 + 1.092975208

Reorder the terms:
1.092975208 + -2.090909091x + x2 = 1.909090909 + 1.092975208

Combine like terms: 1.909090909 + 1.092975208 = 3.002066117
1.092975208 + -2.090909091x + x2 = 3.002066117

Factor a perfect square on the left side:
(x + -1.045454546)(x + -1.045454546) = 3.002066117

Calculate the square root of the right side: 1.732647142

Break this problem into two subproblems by setting 
(x + -1.045454546) equal to 1.732647142 and -1.732647142.

Subproblem 1

x + -1.045454546 = 1.732647142 Simplifying x + -1.045454546 = 1.732647142 Reorder the terms: -1.045454546 + x = 1.732647142 Solving -1.045454546 + x = 1.732647142 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 = 1.732647142 + 1.045454546 Combine like terms: -1.045454546 + 1.045454546 = 0.000000000 0.000000000 + x = 1.732647142 + 1.045454546 x = 1.732647142 + 1.045454546 Combine like terms: 1.732647142 + 1.045454546 = 2.778101688 x = 2.778101688 Simplifying x = 2.778101688

Subproblem 2

x + -1.045454546 = -1.732647142 Simplifying x + -1.045454546 = -1.732647142 Reorder the terms: -1.045454546 + x = -1.732647142 Solving -1.045454546 + x = -1.732647142 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 = -1.732647142 + 1.045454546 Combine like terms: -1.045454546 + 1.045454546 = 0.000000000 0.000000000 + x = -1.732647142 + 1.045454546 x = -1.732647142 + 1.045454546 Combine like terms: -1.732647142 + 1.045454546 = -0.687192596 x = -0.687192596 Simplifying x = -0.687192596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.778101688, -0.687192596}

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