4y(y+7)=21

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Solution for 4y(y+7)=21 equation: 


Simplifying
4y(y + 7) = 21

Reorder the terms:
4y(7 + y) = 21
(7 * 4y + y * 4y) = 21
(28y + 4y2) = 21

Solving
28y + 4y2 = 21

Solving for variable 'y'.

Reorder the terms:
-21 + 28y + 4y2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 28y + 4y2 = 0

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

Divide each side by '4'.
-5.25 + 7y + y2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + 7y + 5.25 + y2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + 7y + y2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + 7y + y2 = 0 + 5.25
7y + y2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
7y + y2 = 5.25

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 = 5.25 + 12.25

Reorder the terms:
12.25 + 7y + y2 = 5.25 + 12.25

Combine like terms: 5.25 + 12.25 = 17.5
12.25 + 7y + y2 = 17.5

Factor a perfect square on the left side:
(y + 3.5)(y + 3.5) = 17.5

Calculate the square root of the right side: 4.183300133

Break this problem into two subproblems by setting 
(y + 3.5) equal to 4.183300133 and -4.183300133.


Subproblem 1
y + 3.5 = 4.183300133

Simplifying
y + 3.5 = 4.183300133

Reorder the terms:
3.5 + y = 4.183300133

Solving
3.5 + y = 4.183300133

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 = 4.183300133 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = 4.183300133 + -3.5
y = 4.183300133 + -3.5

Combine like terms: 4.183300133 + -3.5 = 0.683300133
y = 0.683300133

Simplifying
y = 0.683300133


Subproblem 2
y + 3.5 = -4.183300133

Simplifying
y + 3.5 = -4.183300133

Reorder the terms:
3.5 + y = -4.183300133

Solving
3.5 + y = -4.183300133

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 = -4.183300133 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = -4.183300133 + -3.5
y = -4.183300133 + -3.5

Combine like terms: -4.183300133 + -3.5 = -7.683300133
y = -7.683300133

Simplifying
y = -7.683300133


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.683300133, -7.683300133}

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