4y(y+7)=42

Solution for 4y(y+7)=42 equation:

Simplifying
4y(y + 7) = 42

Reorder the terms:
4y(7 + y) = 42
(7 * 4y + y * 4y) = 42
(28y + 4y2) = 42

Solving
28y + 4y2 = 42

Solving for variable 'y'.

Reorder the terms:
-42 + 28y + 4y2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + 28y + 4y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + 14y + 2y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + 14y + 2y2)' equal to zero and attempt to solve:

Simplifying
-21 + 14y + 2y2 = 0

Solving
-21 + 14y + 2y2 = 0

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

Divide each side by '2'.
-10.5 + 7y + y2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 7y + 10.5 + y2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 7y + y2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 7y + y2 = 0 + 10.5
7y + y2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
7y + y2 = 10.5

The y term is 7y.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7y + 12.25 + y2 = 10.5 + 12.25

Reorder the terms:
12.25 + 7y + y2 = 10.5 + 12.25

Combine like terms: 10.5 + 12.25 = 22.75
12.25 + 7y + y2 = 22.75

Factor a perfect square on the left side:
(y + 3.5)(y + 3.5) = 22.75

Calculate the square root of the right side: 4.769696007

Break this problem into two subproblems by setting 
(y + 3.5) equal to 4.769696007 and -4.769696007.

Subproblem 1
y + 3.5 = 4.769696007

Simplifying
y + 3.5 = 4.769696007

Reorder the terms:
3.5 + y = 4.769696007

Solving
3.5 + y = 4.769696007

Solving for variable 'y'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + y = 4.769696007 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = 4.769696007 + -3.5
y = 4.769696007 + -3.5

Combine like terms: 4.769696007 + -3.5 = 1.269696007
y = 1.269696007

Simplifying
y = 1.269696007

Subproblem 2
y + 3.5 = -4.769696007

Simplifying
y + 3.5 = -4.769696007

Reorder the terms:
3.5 + y = -4.769696007

Solving
3.5 + y = -4.769696007

Solving for variable 'y'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + y = -4.769696007 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = -4.769696007 + -3.5
y = -4.769696007 + -3.5

Combine like terms: -4.769696007 + -3.5 = -8.269696007
y = -8.269696007

Simplifying
y = -8.269696007

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.269696007, -8.269696007}
Solution
y = {1.269696007, -8.269696007}