27x^2-78x-8=0

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Solution for 27x^2-78x-8=0 equation:


Simplifying
27x2 + -78x + -8 = 0

Reorder the terms:
-8 + -78x + 27x2 = 0

Solving
-8 + -78x + 27x2 = 0

Solving for variable 'x'.

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

Divide each side by '27'.
-0.2962962963 + -2.888888889x + x2 = 0

Move the constant term to the right:

Add '0.2962962963' to each side of the equation.
-0.2962962963 + -2.888888889x + 0.2962962963 + x2 = 0 + 0.2962962963

Reorder the terms:
-0.2962962963 + 0.2962962963 + -2.888888889x + x2 = 0 + 0.2962962963

Combine like terms: -0.2962962963 + 0.2962962963 = 0.0000000000
0.0000000000 + -2.888888889x + x2 = 0 + 0.2962962963
-2.888888889x + x2 = 0 + 0.2962962963

Combine like terms: 0 + 0.2962962963 = 0.2962962963
-2.888888889x + x2 = 0.2962962963

The x term is -2.888888889x.  Take half its coefficient (-1.444444445).
Square it (2.086419755) and add it to both sides.

Add '2.086419755' to each side of the equation.
-2.888888889x + 2.086419755 + x2 = 0.2962962963 + 2.086419755

Reorder the terms:
2.086419755 + -2.888888889x + x2 = 0.2962962963 + 2.086419755

Combine like terms: 0.2962962963 + 2.086419755 = 2.3827160513
2.086419755 + -2.888888889x + x2 = 2.3827160513

Factor a perfect square on the left side:
(x + -1.444444445)(x + -1.444444445) = 2.3827160513

Calculate the square root of the right side: 1.543604888

Break this problem into two subproblems by setting 
(x + -1.444444445) equal to 1.543604888 and -1.543604888.

Subproblem 1

x + -1.444444445 = 1.543604888 Simplifying x + -1.444444445 = 1.543604888 Reorder the terms: -1.444444445 + x = 1.543604888 Solving -1.444444445 + x = 1.543604888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + x = 1.543604888 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + x = 1.543604888 + 1.444444445 x = 1.543604888 + 1.444444445 Combine like terms: 1.543604888 + 1.444444445 = 2.988049333 x = 2.988049333 Simplifying x = 2.988049333

Subproblem 2

x + -1.444444445 = -1.543604888 Simplifying x + -1.444444445 = -1.543604888 Reorder the terms: -1.444444445 + x = -1.543604888 Solving -1.444444445 + x = -1.543604888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + x = -1.543604888 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + x = -1.543604888 + 1.444444445 x = -1.543604888 + 1.444444445 Combine like terms: -1.543604888 + 1.444444445 = -0.099160443 x = -0.099160443 Simplifying x = -0.099160443

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.988049333, -0.099160443}

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