3x^2-5x-16=0

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Solution for 3x^2-5x-16=0 equation: 


Simplifying
3x2 + -5x + -16 = 0

Reorder the terms:
-16 + -5x + 3x2 = 0

Solving
-16 + -5x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-5.333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + -1.666666667x + 5.333333333 + x2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + -1.666666667x + x2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 5.333333333
-1.666666667x + x2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
-1.666666667x + x2 = 5.333333333

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 5.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 5.333333333 + 0.6944444447

Combine like terms: 5.333333333 + 0.6944444447 = 6.0277777777
0.6944444447 + -1.666666667x + x2 = 6.0277777777

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 6.0277777777

Calculate the square root of the right side: 2.45515331

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.45515331 and -2.45515331.


Subproblem 1
x + -0.8333333335 = 2.45515331

Simplifying
x + -0.8333333335 = 2.45515331

Reorder the terms:
-0.8333333335 + x = 2.45515331

Solving
-0.8333333335 + x = 2.45515331

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = 2.45515331 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 2.45515331 + 0.8333333335
x = 2.45515331 + 0.8333333335

Combine like terms: 2.45515331 + 0.8333333335 = 3.2884866435
x = 3.2884866435

Simplifying
x = 3.2884866435


Subproblem 2
x + -0.8333333335 = -2.45515331

Simplifying
x + -0.8333333335 = -2.45515331

Reorder the terms:
-0.8333333335 + x = -2.45515331

Solving
-0.8333333335 + x = -2.45515331

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = -2.45515331 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -2.45515331 + 0.8333333335
x = -2.45515331 + 0.8333333335

Combine like terms: -2.45515331 + 0.8333333335 = -1.6218199765
x = -1.6218199765

Simplifying
x = -1.6218199765


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.2884866435, -1.6218199765}

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