6x^2-5x-4=2x+3

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Solution for 6x^2-5x-4=2x+3 equation:


Simplifying
6x2 + -5x + -4 = 2x + 3

Reorder the terms:
-4 + -5x + 6x2 = 2x + 3

Reorder the terms:
-4 + -5x + 6x2 = 3 + 2x

Solving
-4 + -5x + 6x2 = 3 + 2x

Solving for variable 'x'.

Reorder the terms:
-4 + -3 + -5x + -2x + 6x2 = 3 + 2x + -3 + -2x

Combine like terms: -4 + -3 = -7
-7 + -5x + -2x + 6x2 = 3 + 2x + -3 + -2x

Combine like terms: -5x + -2x = -7x
-7 + -7x + 6x2 = 3 + 2x + -3 + -2x

Reorder the terms:
-7 + -7x + 6x2 = 3 + -3 + 2x + -2x

Combine like terms: 3 + -3 = 0
-7 + -7x + 6x2 = 0 + 2x + -2x
-7 + -7x + 6x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-7 + -7x + 6x2 = 0

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

Divide each side by '6'.
-1.166666667 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -1.166666667x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -1.166666667x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -1.166666667x + x2 = 0 + 1.166666667
-1.166666667x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-1.166666667x + x2 = 1.166666667

The x term is -1.166666667x.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667x + 0.3402777780 + x2 = 1.166666667 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 1.166666667 + 0.3402777780

Combine like terms: 1.166666667 + 0.3402777780 = 1.506944445
0.3402777780 + -1.166666667x + x2 = 1.506944445

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 1.506944445

Calculate the square root of the right side: 1.227576655

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 1.227576655 and -1.227576655.

Subproblem 1

x + -0.5833333335 = 1.227576655 Simplifying x + -0.5833333335 = 1.227576655 Reorder the terms: -0.5833333335 + x = 1.227576655 Solving -0.5833333335 + x = 1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = 1.227576655 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.227576655 + 0.5833333335 x = 1.227576655 + 0.5833333335 Combine like terms: 1.227576655 + 0.5833333335 = 1.8109099885 x = 1.8109099885 Simplifying x = 1.8109099885

Subproblem 2

x + -0.5833333335 = -1.227576655 Simplifying x + -0.5833333335 = -1.227576655 Reorder the terms: -0.5833333335 + x = -1.227576655 Solving -0.5833333335 + x = -1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = -1.227576655 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.227576655 + 0.5833333335 x = -1.227576655 + 0.5833333335 Combine like terms: -1.227576655 + 0.5833333335 = -0.6442433215 x = -0.6442433215 Simplifying x = -0.6442433215

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.8109099885, -0.6442433215}

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