8x^2-27x-5=0

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Solution for 8x^2-27x-5=0 equation:


Simplifying
8x2 + -27x + -5 = 0

Reorder the terms:
-5 + -27x + 8x2 = 0

Solving
-5 + -27x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
-0.625 + -3.375x + x2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + -3.375x + 0.625 + x2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + -3.375x + x2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + -3.375x + x2 = 0 + 0.625
-3.375x + x2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
-3.375x + x2 = 0.625

The x term is -3.375x.  Take half its coefficient (-1.6875).
Square it (2.84765625) and add it to both sides.

Add '2.84765625' to each side of the equation.
-3.375x + 2.84765625 + x2 = 0.625 + 2.84765625

Reorder the terms:
2.84765625 + -3.375x + x2 = 0.625 + 2.84765625

Combine like terms: 0.625 + 2.84765625 = 3.47265625
2.84765625 + -3.375x + x2 = 3.47265625

Factor a perfect square on the left side:
(x + -1.6875)(x + -1.6875) = 3.47265625

Calculate the square root of the right side: 1.863506439

Break this problem into two subproblems by setting 
(x + -1.6875) equal to 1.863506439 and -1.863506439.

Subproblem 1

x + -1.6875 = 1.863506439 Simplifying x + -1.6875 = 1.863506439 Reorder the terms: -1.6875 + x = 1.863506439 Solving -1.6875 + x = 1.863506439 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = 1.863506439 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = 1.863506439 + 1.6875 x = 1.863506439 + 1.6875 Combine like terms: 1.863506439 + 1.6875 = 3.551006439 x = 3.551006439 Simplifying x = 3.551006439

Subproblem 2

x + -1.6875 = -1.863506439 Simplifying x + -1.6875 = -1.863506439 Reorder the terms: -1.6875 + x = -1.863506439 Solving -1.6875 + x = -1.863506439 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = -1.863506439 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = -1.863506439 + 1.6875 x = -1.863506439 + 1.6875 Combine like terms: -1.863506439 + 1.6875 = -0.176006439 x = -0.176006439 Simplifying x = -0.176006439

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.551006439, -0.176006439}

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