11x^2-31x-54=0

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Solution for 11x^2-31x-54=0 equation:


Simplifying
11x2 + -31x + -54 = 0

Reorder the terms:
-54 + -31x + 11x2 = 0

Solving
-54 + -31x + 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'.
-4.909090909 + -2.818181818x + x2 = 0

Move the constant term to the right:

Add '4.909090909' to each side of the equation.
-4.909090909 + -2.818181818x + 4.909090909 + x2 = 0 + 4.909090909

Reorder the terms:
-4.909090909 + 4.909090909 + -2.818181818x + x2 = 0 + 4.909090909

Combine like terms: -4.909090909 + 4.909090909 = 0.000000000
0.000000000 + -2.818181818x + x2 = 0 + 4.909090909
-2.818181818x + x2 = 0 + 4.909090909

Combine like terms: 0 + 4.909090909 = 4.909090909
-2.818181818x + x2 = 4.909090909

The x term is -2.818181818x.  Take half its coefficient (-1.409090909).
Square it (1.985537190) and add it to both sides.

Add '1.985537190' to each side of the equation.
-2.818181818x + 1.985537190 + x2 = 4.909090909 + 1.985537190

Reorder the terms:
1.985537190 + -2.818181818x + x2 = 4.909090909 + 1.985537190

Combine like terms: 4.909090909 + 1.985537190 = 6.894628099
1.985537190 + -2.818181818x + x2 = 6.894628099

Factor a perfect square on the left side:
(x + -1.409090909)(x + -1.409090909) = 6.894628099

Calculate the square root of the right side: 2.625762384

Break this problem into two subproblems by setting 
(x + -1.409090909) equal to 2.625762384 and -2.625762384.

Subproblem 1

x + -1.409090909 = 2.625762384 Simplifying x + -1.409090909 = 2.625762384 Reorder the terms: -1.409090909 + x = 2.625762384 Solving -1.409090909 + x = 2.625762384 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.409090909' to each side of the equation. -1.409090909 + 1.409090909 + x = 2.625762384 + 1.409090909 Combine like terms: -1.409090909 + 1.409090909 = 0.000000000 0.000000000 + x = 2.625762384 + 1.409090909 x = 2.625762384 + 1.409090909 Combine like terms: 2.625762384 + 1.409090909 = 4.034853293 x = 4.034853293 Simplifying x = 4.034853293

Subproblem 2

x + -1.409090909 = -2.625762384 Simplifying x + -1.409090909 = -2.625762384 Reorder the terms: -1.409090909 + x = -2.625762384 Solving -1.409090909 + x = -2.625762384 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.409090909' to each side of the equation. -1.409090909 + 1.409090909 + x = -2.625762384 + 1.409090909 Combine like terms: -1.409090909 + 1.409090909 = 0.000000000 0.000000000 + x = -2.625762384 + 1.409090909 x = -2.625762384 + 1.409090909 Combine like terms: -2.625762384 + 1.409090909 = -1.216671475 x = -1.216671475 Simplifying x = -1.216671475

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.034853293, -1.216671475}

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