6x^2+27x-5=0

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Solution for 6x^2+27x-5=0 equation:


Simplifying
6x2 + 27x + -5 = 0

Reorder the terms:
-5 + 27x + 6x2 = 0

Solving
-5 + 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'.
-0.8333333333 + 4.5x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 4.5x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 4.5x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 4.5x + x2 = 0 + 0.8333333333
4.5x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
4.5x + x2 = 0.8333333333

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 = 0.8333333333 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 0.8333333333 + 5.0625

Combine like terms: 0.8333333333 + 5.0625 = 5.8958333333
5.0625 + 4.5x + x2 = 5.8958333333

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 5.8958333333

Calculate the square root of the right side: 2.428133714

Break this problem into two subproblems by setting 
(x + 2.25) equal to 2.428133714 and -2.428133714.

Subproblem 1

x + 2.25 = 2.428133714 Simplifying x + 2.25 = 2.428133714 Reorder the terms: 2.25 + x = 2.428133714 Solving 2.25 + x = 2.428133714 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 = 2.428133714 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.428133714 + -2.25 x = 2.428133714 + -2.25 Combine like terms: 2.428133714 + -2.25 = 0.178133714 x = 0.178133714 Simplifying x = 0.178133714

Subproblem 2

x + 2.25 = -2.428133714 Simplifying x + 2.25 = -2.428133714 Reorder the terms: 2.25 + x = -2.428133714 Solving 2.25 + x = -2.428133714 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 = -2.428133714 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.428133714 + -2.25 x = -2.428133714 + -2.25 Combine like terms: -2.428133714 + -2.25 = -4.678133714 x = -4.678133714 Simplifying x = -4.678133714

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.178133714, -4.678133714}

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