6x^2-27x-20=0

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Solution for 6x^2-27x-20=0 equation:


Simplifying
6x2 + -27x + -20 = 0

Reorder the terms:
-20 + -27x + 6x2 = 0

Solving
-20 + -27x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.333333333 + -4.5x + x2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -4.5x + 3.333333333 + x2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -4.5x + x2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -4.5x + x2 = 0 + 3.333333333
-4.5x + x2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-4.5x + x2 = 3.333333333

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 3.333333333 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 3.333333333 + 5.0625

Combine like terms: 3.333333333 + 5.0625 = 8.395833333
5.0625 + -4.5x + x2 = 8.395833333

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 8.395833333

Calculate the square root of the right side: 2.897556442

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.897556442 and -2.897556442.

Subproblem 1

x + -2.25 = 2.897556442 Simplifying x + -2.25 = 2.897556442 Reorder the terms: -2.25 + x = 2.897556442 Solving -2.25 + x = 2.897556442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 2.897556442 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.897556442 + 2.25 x = 2.897556442 + 2.25 Combine like terms: 2.897556442 + 2.25 = 5.147556442 x = 5.147556442 Simplifying x = 5.147556442

Subproblem 2

x + -2.25 = -2.897556442 Simplifying x + -2.25 = -2.897556442 Reorder the terms: -2.25 + x = -2.897556442 Solving -2.25 + x = -2.897556442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -2.897556442 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.897556442 + 2.25 x = -2.897556442 + 2.25 Combine like terms: -2.897556442 + 2.25 = -0.647556442 x = -0.647556442 Simplifying x = -0.647556442

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.147556442, -0.647556442}

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