1.5x^2-5x-6=0

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Solution for 1.5x^2-5x-6=0 equation: 


Simplifying
1.5x2 + -5x + -6 = 0

Reorder the terms:
-6 + -5x + 1.5x2 = 0

Solving
-6 + -5x + 1.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.5'.
-4 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -3.333333333x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -3.333333333x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -3.333333333x + x2 = 0 + 4
-3.333333333x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-3.333333333x + x2 = 4

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = 4 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = 4 + 2.777777779

Combine like terms: 4 + 2.777777779 = 6.777777779
2.777777779 + -3.333333333x + x2 = 6.777777779

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 6.777777779

Calculate the square root of the right side: 2.603416559

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 2.603416559 and -2.603416559.


Subproblem 1
x + -1.666666667 = 2.603416559

Simplifying
x + -1.666666667 = 2.603416559

Reorder the terms:
-1.666666667 + x = 2.603416559

Solving
-1.666666667 + x = 2.603416559

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = 2.603416559 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = 2.603416559 + 1.666666667
x = 2.603416559 + 1.666666667

Combine like terms: 2.603416559 + 1.666666667 = 4.270083226
x = 4.270083226

Simplifying
x = 4.270083226


Subproblem 2
x + -1.666666667 = -2.603416559

Simplifying
x + -1.666666667 = -2.603416559

Reorder the terms:
-1.666666667 + x = -2.603416559

Solving
-1.666666667 + x = -2.603416559

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = -2.603416559 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = -2.603416559 + 1.666666667
x = -2.603416559 + 1.666666667

Combine like terms: -2.603416559 + 1.666666667 = -0.936749892
x = -0.936749892

Simplifying
x = -0.936749892


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.270083226, -0.936749892}

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