6x=32x^2-48x-81

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Solution for 6x=32x^2-48x-81 equation:


Simplifying
6x = 32x2 + -48x + -81

Reorder the terms:
6x = -81 + -48x + 32x2

Solving
6x = -81 + -48x + 32x2

Solving for variable 'x'.

Reorder the terms:
81 + 6x + 48x + -32x2 = -81 + -48x + 32x2 + 81 + 48x + -32x2

Combine like terms: 6x + 48x = 54x
81 + 54x + -32x2 = -81 + -48x + 32x2 + 81 + 48x + -32x2

Reorder the terms:
81 + 54x + -32x2 = -81 + 81 + -48x + 48x + 32x2 + -32x2

Combine like terms: -81 + 81 = 0
81 + 54x + -32x2 = 0 + -48x + 48x + 32x2 + -32x2
81 + 54x + -32x2 = -48x + 48x + 32x2 + -32x2

Combine like terms: -48x + 48x = 0
81 + 54x + -32x2 = 0 + 32x2 + -32x2
81 + 54x + -32x2 = 32x2 + -32x2

Combine like terms: 32x2 + -32x2 = 0
81 + 54x + -32x2 = 0

Begin completing the square.  Divide all terms by
-32 the coefficient of the squared term: 

Divide each side by '-32'.
-2.53125 + -1.6875x + x2 = 0

Move the constant term to the right:

Add '2.53125' to each side of the equation.
-2.53125 + -1.6875x + 2.53125 + x2 = 0 + 2.53125

Reorder the terms:
-2.53125 + 2.53125 + -1.6875x + x2 = 0 + 2.53125

Combine like terms: -2.53125 + 2.53125 = 0.00000
0.00000 + -1.6875x + x2 = 0 + 2.53125
-1.6875x + x2 = 0 + 2.53125

Combine like terms: 0 + 2.53125 = 2.53125
-1.6875x + x2 = 2.53125

The x term is -1.6875x.  Take half its coefficient (-0.84375).
Square it (0.7119140625) and add it to both sides.

Add '0.7119140625' to each side of the equation.
-1.6875x + 0.7119140625 + x2 = 2.53125 + 0.7119140625

Reorder the terms:
0.7119140625 + -1.6875x + x2 = 2.53125 + 0.7119140625

Combine like terms: 2.53125 + 0.7119140625 = 3.2431640625
0.7119140625 + -1.6875x + x2 = 3.2431640625

Factor a perfect square on the left side:
(x + -0.84375)(x + -0.84375) = 3.2431640625

Calculate the square root of the right side: 1.800878692

Break this problem into two subproblems by setting 
(x + -0.84375) equal to 1.800878692 and -1.800878692.

Subproblem 1

x + -0.84375 = 1.800878692 Simplifying x + -0.84375 = 1.800878692 Reorder the terms: -0.84375 + x = 1.800878692 Solving -0.84375 + x = 1.800878692 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = 1.800878692 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = 1.800878692 + 0.84375 x = 1.800878692 + 0.84375 Combine like terms: 1.800878692 + 0.84375 = 2.644628692 x = 2.644628692 Simplifying x = 2.644628692

Subproblem 2

x + -0.84375 = -1.800878692 Simplifying x + -0.84375 = -1.800878692 Reorder the terms: -0.84375 + x = -1.800878692 Solving -0.84375 + x = -1.800878692 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = -1.800878692 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = -1.800878692 + 0.84375 x = -1.800878692 + 0.84375 Combine like terms: -1.800878692 + 0.84375 = -0.957128692 x = -0.957128692 Simplifying x = -0.957128692

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.644628692, -0.957128692}

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