6x^2-7x-16=0

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Solution for 6x^2-7x-16=0 equation:


Simplifying
6x2 + -7x + -16 = 0

Reorder the terms:
-16 + -7x + 6x2 = 0

Solving
-16 + -7x + 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'.
-2.666666667 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -1.166666667x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -1.166666667x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -1.166666667x + x2 = 0 + 2.666666667
-1.166666667x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-1.166666667x + x2 = 2.666666667

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 = 2.666666667 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 2.666666667 + 0.3402777780

Combine like terms: 2.666666667 + 0.3402777780 = 3.006944445
0.3402777780 + -1.166666667x + x2 = 3.006944445

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 3.006944445

Calculate the square root of the right side: 1.734054337

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 1.734054337 and -1.734054337.

Subproblem 1

x + -0.5833333335 = 1.734054337 Simplifying x + -0.5833333335 = 1.734054337 Reorder the terms: -0.5833333335 + x = 1.734054337 Solving -0.5833333335 + x = 1.734054337 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 = 1.734054337 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.734054337 + 0.5833333335 x = 1.734054337 + 0.5833333335 Combine like terms: 1.734054337 + 0.5833333335 = 2.3173876705 x = 2.3173876705 Simplifying x = 2.3173876705

Subproblem 2

x + -0.5833333335 = -1.734054337 Simplifying x + -0.5833333335 = -1.734054337 Reorder the terms: -0.5833333335 + x = -1.734054337 Solving -0.5833333335 + x = -1.734054337 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 = -1.734054337 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.734054337 + 0.5833333335 x = -1.734054337 + 0.5833333335 Combine like terms: -1.734054337 + 0.5833333335 = -1.1507210035 x = -1.1507210035 Simplifying x = -1.1507210035

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.3173876705, -1.1507210035}

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