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 = 18.02777778
10.02777778 + 6.333333333y + y2 = 18.02777778

Factor a perfect square on the left side:
(y + 3.166666667)(y + 3.166666667) = 18.02777778

Calculate the square root of the right side: 4.245913068

Break this problem into two subproblems by setting 
(y + 3.166666667) equal to 4.245913068 and -4.245913068.


Subproblem 1
y + 3.166666667 = 4.245913068

Simplifying
y + 3.166666667 = 4.245913068

Reorder the terms:
3.166666667 + y = 4.245913068

Solving
3.166666667 + y = 4.245913068

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 = 4.245913068 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + y = 4.245913068 + -3.166666667
y = 4.245913068 + -3.166666667

Combine like terms: 4.245913068 + -3.166666667 = 1.079246401
y = 1.079246401

Simplifying
y = 1.079246401


Subproblem 2
y + 3.166666667 = -4.245913068

Simplifying
y + 3.166666667 = -4.245913068

Reorder the terms:
3.166666667 + y = -4.245913068

Solving
3.166666667 + y = -4.245913068

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 = -4.245913068 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + y = -4.245913068 + -3.166666667
y = -4.245913068 + -3.166666667

Combine like terms: -4.245913068 + -3.166666667 = -7.412579735
y = -7.412579735

Simplifying
y = -7.412579735


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.079246401, -7.412579735}

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