6x^2+1.5x-27=0

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Solution for 6x^2+1.5x-27=0 equation:


Simplifying
6x2 + 1.5x + -27 = 0

Reorder the terms:
-27 + 1.5x + 6x2 = 0

Solving
-27 + 1.5x + 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'.
-4.5 + 0.25x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 0.25x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 0.25x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 0.25x + x2 = 0 + 4.5
0.25x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
0.25x + x2 = 4.5

The x term is 0.25x.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25x + 0.015625 + x2 = 4.5 + 0.015625

Reorder the terms:
0.015625 + 0.25x + x2 = 4.5 + 0.015625

Combine like terms: 4.5 + 0.015625 = 4.515625
0.015625 + 0.25x + x2 = 4.515625

Factor a perfect square on the left side:
(x + 0.125)(x + 0.125) = 4.515625

Calculate the square root of the right side: 2.125

Break this problem into two subproblems by setting 
(x + 0.125) equal to 2.125 and -2.125.

Subproblem 1

x + 0.125 = 2.125 Simplifying x + 0.125 = 2.125 Reorder the terms: 0.125 + x = 2.125 Solving 0.125 + x = 2.125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 2.125 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 2.125 + -0.125 x = 2.125 + -0.125 Combine like terms: 2.125 + -0.125 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + 0.125 = -2.125 Simplifying x + 0.125 = -2.125 Reorder the terms: 0.125 + x = -2.125 Solving 0.125 + x = -2.125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -2.125 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -2.125 + -0.125 x = -2.125 + -0.125 Combine like terms: -2.125 + -0.125 = -2.25 x = -2.25 Simplifying x = -2.25

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, -2.25}

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