9x^2-54x-1=0

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Solution for 9x^2-54x-1=0 equation:


Simplifying
9x2 + -54x + -1 = 0

Reorder the terms:
-1 + -54x + 9x2 = 0

Solving
-1 + -54x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-0.1111111111 + -6x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -6x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -6x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -6x + x2 = 0 + 0.1111111111
-6x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-6x + x2 = 0.1111111111

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = 0.1111111111 + 9

Reorder the terms:
9 + -6x + x2 = 0.1111111111 + 9

Combine like terms: 0.1111111111 + 9 = 9.1111111111
9 + -6x + x2 = 9.1111111111

Factor a perfect square on the left side:
(x + -3)(x + -3) = 9.1111111111

Calculate the square root of the right side: 3.018461713

Break this problem into two subproblems by setting 
(x + -3) equal to 3.018461713 and -3.018461713.

Subproblem 1

x + -3 = 3.018461713 Simplifying x + -3 = 3.018461713 Reorder the terms: -3 + x = 3.018461713 Solving -3 + x = 3.018461713 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 3.018461713 + 3 Combine like terms: -3 + 3 = 0 0 + x = 3.018461713 + 3 x = 3.018461713 + 3 Combine like terms: 3.018461713 + 3 = 6.018461713 x = 6.018461713 Simplifying x = 6.018461713

Subproblem 2

x + -3 = -3.018461713 Simplifying x + -3 = -3.018461713 Reorder the terms: -3 + x = -3.018461713 Solving -3 + x = -3.018461713 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -3.018461713 + 3 Combine like terms: -3 + 3 = 0 0 + x = -3.018461713 + 3 x = -3.018461713 + 3 Combine like terms: -3.018461713 + 3 = -0.018461713 x = -0.018461713 Simplifying x = -0.018461713

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.018461713, -0.018461713}

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