3x^2+(19+1)x+24=0

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Solution for 3x^2+(19+1)x+24=0 equation: 


Simplifying
3x2 + (19 + 1) * x + 24 = 0

Combine like terms: 19 + 1 = 20
3x2 + (20) * x + 24 = 0

Reorder the terms:
24 + 20x + 3x2 = 0

Solving
24 + 20x + 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'.
8 + 6.666666667x + x2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 6.666666667x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 6.666666667x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 6.666666667x + x2 = 0 + -8
6.666666667x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
6.666666667x + x2 = -8

The x term is 6.666666667x.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667x + 11.11111112 + x2 = -8 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667x + x2 = -8 + 11.11111112

Combine like terms: -8 + 11.11111112 = 3.11111112
11.11111112 + 6.666666667x + x2 = 3.11111112

Factor a perfect square on the left side:
(x + 3.333333334)(x + 3.333333334) = 3.11111112

Calculate the square root of the right side: 1.76383421

Break this problem into two subproblems by setting 
(x + 3.333333334) equal to 1.76383421 and -1.76383421.


Subproblem 1
x + 3.333333334 = 1.76383421

Simplifying
x + 3.333333334 = 1.76383421

Reorder the terms:
3.333333334 + x = 1.76383421

Solving
3.333333334 + x = 1.76383421

Solving for variable 'x'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + x = 1.76383421 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = 1.76383421 + -3.333333334
x = 1.76383421 + -3.333333334

Combine like terms: 1.76383421 + -3.333333334 = -1.569499124
x = -1.569499124

Simplifying
x = -1.569499124


Subproblem 2
x + 3.333333334 = -1.76383421

Simplifying
x + 3.333333334 = -1.76383421

Reorder the terms:
3.333333334 + x = -1.76383421

Solving
3.333333334 + x = -1.76383421

Solving for variable 'x'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + x = -1.76383421 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = -1.76383421 + -3.333333334
x = -1.76383421 + -3.333333334

Combine like terms: -1.76383421 + -3.333333334 = -5.097167544
x = -5.097167544

Simplifying
x = -5.097167544


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.569499124, -5.097167544}

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