4y^2+6y-84=0

Solution for 4y^2+6y-84=0 equation:

Simplifying
4y2 + 6y + -84 = 0

Reorder the terms:
-84 + 6y + 4y2 = 0

Solving
-84 + 6y + 4y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 3y + 2y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-42 + 3y + 2y2)' equal to zero and attempt to solve:

Simplifying
-42 + 3y + 2y2 = 0

Solving
-42 + 3y + 2y2 = 0

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

Divide each side by '2'.
-21 + 1.5y + y2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 1.5y + 21 + y2 = 0 + 21

Reorder the terms:
-21 + 21 + 1.5y + y2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 1.5y + y2 = 0 + 21
1.5y + y2 = 0 + 21

Combine like terms: 0 + 21 = 21
1.5y + y2 = 21

The y term is 1.5y.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5y + 0.5625 + y2 = 21 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 21 + 0.5625

Combine like terms: 21 + 0.5625 = 21.5625
0.5625 + 1.5y + y2 = 21.5625

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 21.5625

Calculate the square root of the right side: 4.643543905

Break this problem into two subproblems by setting 
(y + 0.75) equal to 4.643543905 and -4.643543905.

Subproblem 1
y + 0.75 = 4.643543905

Simplifying
y + 0.75 = 4.643543905

Reorder the terms:
0.75 + y = 4.643543905

Solving
0.75 + y = 4.643543905

Solving for variable 'y'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + y = 4.643543905 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + y = 4.643543905 + -0.75
y = 4.643543905 + -0.75

Combine like terms: 4.643543905 + -0.75 = 3.893543905
y = 3.893543905

Simplifying
y = 3.893543905

Subproblem 2
y + 0.75 = -4.643543905

Simplifying
y + 0.75 = -4.643543905

Reorder the terms:
0.75 + y = -4.643543905

Solving
0.75 + y = -4.643543905

Solving for variable 'y'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + y = -4.643543905 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + y = -4.643543905 + -0.75
y = -4.643543905 + -0.75

Combine like terms: -4.643543905 + -0.75 = -5.393543905
y = -5.393543905

Simplifying
y = -5.393543905

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.893543905, -5.393543905}
Solution
y = {3.893543905, -5.393543905}