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 = 4.590667292
x = 4.590667292

Simplifying
x = 4.590667292


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 = 1.742666042
x = 1.742666042

Simplifying
x = 1.742666042


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

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