54x^2-11x-7=0

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Solution for 54x^2-11x-7=0 equation:


Simplifying
54x2 + -11x + -7 = 0

Reorder the terms:
-7 + -11x + 54x2 = 0

Solving
-7 + -11x + 54x2 = 0

Solving for variable 'x'.

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

Divide each side by '54'.
-0.1296296296 + -0.2037037037x + x2 = 0

Move the constant term to the right:

Add '0.1296296296' to each side of the equation.
-0.1296296296 + -0.2037037037x + 0.1296296296 + x2 = 0 + 0.1296296296

Reorder the terms:
-0.1296296296 + 0.1296296296 + -0.2037037037x + x2 = 0 + 0.1296296296

Combine like terms: -0.1296296296 + 0.1296296296 = 0.0000000000
0.0000000000 + -0.2037037037x + x2 = 0 + 0.1296296296
-0.2037037037x + x2 = 0 + 0.1296296296

Combine like terms: 0 + 0.1296296296 = 0.1296296296
-0.2037037037x + x2 = 0.1296296296

The x term is -0.2037037037x.  Take half its coefficient (-0.1018518519).
Square it (0.01037379974) and add it to both sides.

Add '0.01037379974' to each side of the equation.
-0.2037037037x + 0.01037379974 + x2 = 0.1296296296 + 0.01037379974

Reorder the terms:
0.01037379974 + -0.2037037037x + x2 = 0.1296296296 + 0.01037379974

Combine like terms: 0.1296296296 + 0.01037379974 = 0.14000342934
0.01037379974 + -0.2037037037x + x2 = 0.14000342934

Factor a perfect square on the left side:
(x + -0.1018518519)(x + -0.1018518519) = 0.14000342934

Calculate the square root of the right side: 0.374170321

Break this problem into two subproblems by setting 
(x + -0.1018518519) equal to 0.374170321 and -0.374170321.

Subproblem 1

x + -0.1018518519 = 0.374170321 Simplifying x + -0.1018518519 = 0.374170321 Reorder the terms: -0.1018518519 + x = 0.374170321 Solving -0.1018518519 + x = 0.374170321 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1018518519' to each side of the equation. -0.1018518519 + 0.1018518519 + x = 0.374170321 + 0.1018518519 Combine like terms: -0.1018518519 + 0.1018518519 = 0.0000000000 0.0000000000 + x = 0.374170321 + 0.1018518519 x = 0.374170321 + 0.1018518519 Combine like terms: 0.374170321 + 0.1018518519 = 0.4760221729 x = 0.4760221729 Simplifying x = 0.4760221729

Subproblem 2

x + -0.1018518519 = -0.374170321 Simplifying x + -0.1018518519 = -0.374170321 Reorder the terms: -0.1018518519 + x = -0.374170321 Solving -0.1018518519 + x = -0.374170321 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1018518519' to each side of the equation. -0.1018518519 + 0.1018518519 + x = -0.374170321 + 0.1018518519 Combine like terms: -0.1018518519 + 0.1018518519 = 0.0000000000 0.0000000000 + x = -0.374170321 + 0.1018518519 x = -0.374170321 + 0.1018518519 Combine like terms: -0.374170321 + 0.1018518519 = -0.2723184691 x = -0.2723184691 Simplifying x = -0.2723184691

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.4760221729, -0.2723184691}

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