6x^2-x-6=0

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


Simplifying
6x2 + -1x + -6 = 0

Reorder the terms:
-6 + -1x + 6x2 = 0

Solving
-6 + -1x + 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'.
-1 + -0.1666666667x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.1666666667x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.1666666667x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.1666666667x + x2 = 0 + 1
-0.1666666667x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.1666666667x + x2 = 1

The x term is -0.1666666667x.  Take half its coefficient (-0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
-0.1666666667x + 0.006944444447 + x2 = 1 + 0.006944444447

Reorder the terms:
0.006944444447 + -0.1666666667x + x2 = 1 + 0.006944444447

Combine like terms: 1 + 0.006944444447 = 1.006944444447
0.006944444447 + -0.1666666667x + x2 = 1.006944444447

Factor a perfect square on the left side:
(x + -0.08333333335)(x + -0.08333333335) = 1.006944444447

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + -0.08333333335) equal to 1.003466215 and -1.003466215.

Subproblem 1

x + -0.08333333335 = 1.003466215 Simplifying x + -0.08333333335 = 1.003466215 Reorder the terms: -0.08333333335 + x = 1.003466215 Solving -0.08333333335 + x = 1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + x = 1.003466215 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + x = 1.003466215 + 0.08333333335 x = 1.003466215 + 0.08333333335 Combine like terms: 1.003466215 + 0.08333333335 = 1.08679954835 x = 1.08679954835 Simplifying x = 1.08679954835

Subproblem 2

x + -0.08333333335 = -1.003466215 Simplifying x + -0.08333333335 = -1.003466215 Reorder the terms: -0.08333333335 + x = -1.003466215 Solving -0.08333333335 + x = -1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + x = -1.003466215 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + x = -1.003466215 + 0.08333333335 x = -1.003466215 + 0.08333333335 Combine like terms: -1.003466215 + 0.08333333335 = -0.92013288165 x = -0.92013288165 Simplifying x = -0.92013288165

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.08679954835, -0.92013288165}

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