3x^2+27x-6.5=0

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Solution for 3x^2+27x-6.5=0 equation:


Simplifying
3x2 + 27x + -6.5 = 0

Reorder the terms:
-6.5 + 27x + 3x2 = 0

Solving
-6.5 + 27x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-2.166666667 + 9x + x2 = 0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + 9x + 2.166666667 + x2 = 0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + 9x + x2 = 0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + 9x + x2 = 0 + 2.166666667
9x + x2 = 0 + 2.166666667

Combine like terms: 0 + 2.166666667 = 2.166666667
9x + x2 = 2.166666667

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = 2.166666667 + 20.25

Reorder the terms:
20.25 + 9x + x2 = 2.166666667 + 20.25

Combine like terms: 2.166666667 + 20.25 = 22.416666667
20.25 + 9x + x2 = 22.416666667

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 22.416666667

Calculate the square root of the right side: 4.734624237

Break this problem into two subproblems by setting 
(x + 4.5) equal to 4.734624237 and -4.734624237.

Subproblem 1

x + 4.5 = 4.734624237 Simplifying x + 4.5 = 4.734624237 Reorder the terms: 4.5 + x = 4.734624237 Solving 4.5 + x = 4.734624237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = 4.734624237 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = 4.734624237 + -4.5 x = 4.734624237 + -4.5 Combine like terms: 4.734624237 + -4.5 = 0.234624237 x = 0.234624237 Simplifying x = 0.234624237

Subproblem 2

x + 4.5 = -4.734624237 Simplifying x + 4.5 = -4.734624237 Reorder the terms: 4.5 + x = -4.734624237 Solving 4.5 + x = -4.734624237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = -4.734624237 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = -4.734624237 + -4.5 x = -4.734624237 + -4.5 Combine like terms: -4.734624237 + -4.5 = -9.234624237 x = -9.234624237 Simplifying x = -9.234624237

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.234624237, -9.234624237}

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