6x^2-5x-4=1

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Solution for 6x^2-5x-4=1 equation: 


Simplifying
6x2 + -5x + -4 = 1

Reorder the terms:
-4 + -5x + 6x2 = 1

Solving
-4 + -5x + 6x2 = 1

Solving for variable 'x'.

Reorder the terms:
-4 + -1 + -5x + 6x2 = 1 + -1

Combine like terms: -4 + -1 = -5
-5 + -5x + 6x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-5 + -5x + 6x2 = 0

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

Divide each side by '6'.
-0.8333333333 + -0.8333333333x + x2 = 0

Move the constant term to the right:

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

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

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

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

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 0.8333333333 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 0.8333333333 + 0.1736111111

Combine like terms: 0.8333333333 + 0.1736111111 = 1.0069444444
0.1736111111 + -0.8333333333x + x2 = 1.0069444444

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 1.0069444444

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 1.003466215 and -1.003466215.


Subproblem 1
x + -0.4166666667 = 1.003466215

Simplifying
x + -0.4166666667 = 1.003466215

Reorder the terms:
-0.4166666667 + x = 1.003466215

Solving
-0.4166666667 + x = 1.003466215

Solving for variable 'x'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + x = 1.003466215 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + x = 1.003466215 + 0.4166666667
x = 1.003466215 + 0.4166666667

Combine like terms: 1.003466215 + 0.4166666667 = 1.4201328817
x = 1.4201328817

Simplifying
x = 1.4201328817


Subproblem 2
x + -0.4166666667 = -1.003466215

Simplifying
x + -0.4166666667 = -1.003466215

Reorder the terms:
-0.4166666667 + x = -1.003466215

Solving
-0.4166666667 + x = -1.003466215

Solving for variable 'x'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + x = -1.003466215 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + x = -1.003466215 + 0.4166666667
x = -1.003466215 + 0.4166666667

Combine like terms: -1.003466215 + 0.4166666667 = -0.5867995483
x = -0.5867995483

Simplifying
x = -0.5867995483


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4201328817, -0.5867995483}

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