-3y^2+19y-24=0

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


Simplifying
-3y2 + 19y + -24 = 0

Reorder the terms:
-24 + 19y + -3y2 = 0

Solving
-24 + 19y + -3y2 = 0

Solving for variable 'y'.

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

Divide each side by '-3'.
8 + -6.333333333y + y2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -6.333333333y + -8 + y2 = 0 + -8

Reorder the terms:
8 + -8 + -6.333333333y + y2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -6.333333333y + y2 = 0 + -8
-6.333333333y + y2 = 0 + -8

Combine like terms: 0 + -8 = -8
-6.333333333y + y2 = -8

The y term is -6.333333333y.  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.
-6.333333333y + 10.02777778 + y2 = -8 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333y + y2 = -8 + 10.02777778

Combine like terms: -8 + 10.02777778 = 2.02777778
10.02777778 + -6.333333333y + y2 = 2.02777778

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

Calculate the square root of the right side: 1.424000625

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


Subproblem 1
y + -3.166666667 = 1.424000625

Simplifying
y + -3.166666667 = 1.424000625

Reorder the terms:
-3.166666667 + y = 1.424000625

Solving
-3.166666667 + y = 1.424000625

Solving for variable 'y'.

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

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

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

Combine like terms: 1.424000625 + 3.166666667 = 4.590667292
y = 4.590667292

Simplifying
y = 4.590667292


Subproblem 2
y + -3.166666667 = -1.424000625

Simplifying
y + -3.166666667 = -1.424000625

Reorder the terms:
-3.166666667 + y = -1.424000625

Solving
-3.166666667 + y = -1.424000625

Solving for variable 'y'.

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

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

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

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

Simplifying
y = 1.742666042


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

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