3x^2+19x+24=0

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


Simplifying
3x2 + 19x + 24 = 0

Reorder the terms:
24 + 19x + 3x2 = 0

Solving
24 + 19x + 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 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 6.333333333x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 6.333333333x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 6.333333333x + x2 = 0 + -8
6.333333333x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
6.333333333x + x2 = -8

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = -8 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = -8 + 10.02777778

Combine like terms: -8 + 10.02777778 = 2.02777778
10.02777778 + 6.333333333x + x2 = 2.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 2.02777778

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 1.424000625 and -1.424000625.


Subproblem 1
x + 3.166666667 = 1.424000625

Simplifying
x + 3.166666667 = 1.424000625

Reorder the terms:
3.166666667 + x = 1.424000625

Solving
3.166666667 + x = 1.424000625

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = 1.424000625 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = 1.424000625 + -3.166666667
x = 1.424000625 + -3.166666667

Combine like terms: 1.424000625 + -3.166666667 = -1.742666042
x = -1.742666042

Simplifying
x = -1.742666042


Subproblem 2
x + 3.166666667 = -1.424000625

Simplifying
x + 3.166666667 = -1.424000625

Reorder the terms:
3.166666667 + x = -1.424000625

Solving
3.166666667 + x = -1.424000625

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = -1.424000625 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = -1.424000625 + -3.166666667
x = -1.424000625 + -3.166666667

Combine like terms: -1.424000625 + -3.166666667 = -4.590667292
x = -4.590667292

Simplifying
x = -4.590667292


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.742666042, -4.590667292}

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