(1+y)+y(y)=28

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Solution for (1+y)+y(y)=28 equation:


Simplifying
(1 + y) + y(y) = 28

Remove parenthesis around (1 + y)
1 + y + y(y) = 28

Multiply y * y
1 + y + y2 = 28

Solving
1 + y + y2 = 28

Solving for variable 'y'.

Reorder the terms:
1 + -28 + y + y2 = 28 + -28

Combine like terms: 1 + -28 = -27
-27 + y + y2 = 28 + -28

Combine like terms: 28 + -28 = 0
-27 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + y + 27 + y2 = 0 + 27

Reorder the terms:
-27 + 27 + y + y2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + y + y2 = 0 + 27
y + y2 = 0 + 27

Combine like terms: 0 + 27 = 27
y + y2 = 27

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
y + 0.25 + y2 = 27 + 0.25

Reorder the terms:
0.25 + y + y2 = 27 + 0.25

Combine like terms: 27 + 0.25 = 27.25
0.25 + y + y2 = 27.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(y + 0.5) equal to 5.220153254 and -5.220153254.

Subproblem 1

y + 0.5 = 5.220153254 Simplifying y + 0.5 = 5.220153254 Reorder the terms: 0.5 + y = 5.220153254 Solving 0.5 + y = 5.220153254 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 5.220153254 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 5.220153254 + -0.5 y = 5.220153254 + -0.5 Combine like terms: 5.220153254 + -0.5 = 4.720153254 y = 4.720153254 Simplifying y = 4.720153254

Subproblem 2

y + 0.5 = -5.220153254 Simplifying y + 0.5 = -5.220153254 Reorder the terms: 0.5 + y = -5.220153254 Solving 0.5 + y = -5.220153254 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -5.220153254 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -5.220153254 + -0.5 y = -5.220153254 + -0.5 Combine like terms: -5.220153254 + -0.5 = -5.720153254 y = -5.720153254 Simplifying y = -5.720153254

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.720153254, -5.720153254}

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