5(y+1)(y+1)=25

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Solution for 5(y+1)(y+1)=25 equation: 


Simplifying
5(y + 1)(y + 1) = 25

Reorder the terms:
5(1 + y)(y + 1) = 25

Reorder the terms:
5(1 + y)(1 + y) = 25

Multiply (1 + y) * (1 + y)
5(1(1 + y) + y(1 + y)) = 25
5((1 * 1 + y * 1) + y(1 + y)) = 25
5((1 + 1y) + y(1 + y)) = 25
5(1 + 1y + (1 * y + y * y)) = 25
5(1 + 1y + (1y + y2)) = 25

Combine like terms: 1y + 1y = 2y
5(1 + 2y + y2) = 25
(1 * 5 + 2y * 5 + y2 * 5) = 25
(5 + 10y + 5y2) = 25

Solving
5 + 10y + 5y2 = 25

Solving for variable 'y'.

Reorder the terms:
5 + -25 + 10y + 5y2 = 25 + -25

Combine like terms: 5 + -25 = -20
-20 + 10y + 5y2 = 25 + -25

Combine like terms: 25 + -25 = 0
-20 + 10y + 5y2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-4 + 2y + y2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-4 + 2y + y2)' equal to zero and attempt to solve:

Simplifying
-4 + 2y + y2 = 0

Solving
-4 + 2y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + 2y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2y + y2 = 0 + 4
2y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
2y + y2 = 4

The y term is 2y.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2y + 1 + y2 = 4 + 1

Reorder the terms:
1 + 2y + y2 = 4 + 1

Combine like terms: 4 + 1 = 5
1 + 2y + y2 = 5

Factor a perfect square on the left side:
(y + 1)(y + 1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(y + 1) equal to 2.236067978 and -2.236067978.


Subproblem 1
y + 1 = 2.236067978

Simplifying
y + 1 = 2.236067978

Reorder the terms:
1 + y = 2.236067978

Solving
1 + y = 2.236067978

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = 2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + y = 2.236067978 + -1
y = 2.236067978 + -1

Combine like terms: 2.236067978 + -1 = 1.236067978
y = 1.236067978

Simplifying
y = 1.236067978


Subproblem 2
y + 1 = -2.236067978

Simplifying
y + 1 = -2.236067978

Reorder the terms:
1 + y = -2.236067978

Solving
1 + y = -2.236067978

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = -2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + y = -2.236067978 + -1
y = -2.236067978 + -1

Combine like terms: -2.236067978 + -1 = -3.236067978
y = -3.236067978

Simplifying
y = -3.236067978


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.236067978, -3.236067978}
Solution
y = {1.236067978, -3.236067978}

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