18x^2-27x-4=0

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Solution for 18x^2-27x-4=0 equation:


Simplifying
18x2 + -27x + -4 = 0

Reorder the terms:
-4 + -27x + 18x2 = 0

Solving
-4 + -27x + 18x2 = 0

Solving for variable 'x'.

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

Divide each side by '18'.
-0.2222222222 + -1.5x + x2 = 0

Move the constant term to the right:

Add '0.2222222222' to each side of the equation.
-0.2222222222 + -1.5x + 0.2222222222 + x2 = 0 + 0.2222222222

Reorder the terms:
-0.2222222222 + 0.2222222222 + -1.5x + x2 = 0 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + -1.5x + x2 = 0 + 0.2222222222
-1.5x + x2 = 0 + 0.2222222222

Combine like terms: 0 + 0.2222222222 = 0.2222222222
-1.5x + x2 = 0.2222222222

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 0.2222222222 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 0.2222222222 + 0.5625

Combine like terms: 0.2222222222 + 0.5625 = 0.7847222222
0.5625 + -1.5x + x2 = 0.7847222222

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 0.7847222222

Calculate the square root of the right side: 0.885845484

Break this problem into two subproblems by setting 
(x + -0.75) equal to 0.885845484 and -0.885845484.

Subproblem 1

x + -0.75 = 0.885845484 Simplifying x + -0.75 = 0.885845484 Reorder the terms: -0.75 + x = 0.885845484 Solving -0.75 + x = 0.885845484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 0.885845484 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 0.885845484 + 0.75 x = 0.885845484 + 0.75 Combine like terms: 0.885845484 + 0.75 = 1.635845484 x = 1.635845484 Simplifying x = 1.635845484

Subproblem 2

x + -0.75 = -0.885845484 Simplifying x + -0.75 = -0.885845484 Reorder the terms: -0.75 + x = -0.885845484 Solving -0.75 + x = -0.885845484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -0.885845484 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -0.885845484 + 0.75 x = -0.885845484 + 0.75 Combine like terms: -0.885845484 + 0.75 = -0.135845484 x = -0.135845484 Simplifying x = -0.135845484

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.635845484, -0.135845484}

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