Y(8Y+10)=3068

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Solution for Y(8Y+10)=3068 equation: 


Simplifying
Y(8Y + 10) = 3068

Reorder the terms:
Y(10 + 8Y) = 3068
(10 * Y + 8Y * Y) = 3068
(10Y + 8Y2) = 3068

Solving
10Y + 8Y2 = 3068

Solving for variable 'Y'.

Reorder the terms:
-3068 + 10Y + 8Y2 = 3068 + -3068

Combine like terms: 3068 + -3068 = 0
-3068 + 10Y + 8Y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1534 + 5Y + 4Y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-1534 + 5Y + 4Y2)' equal to zero and attempt to solve:

Simplifying
-1534 + 5Y + 4Y2 = 0

Solving
-1534 + 5Y + 4Y2 = 0

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

Divide each side by '4'.
-383.5 + 1.25Y + Y2 = 0

Move the constant term to the right:

Add '383.5' to each side of the equation.
-383.5 + 1.25Y + 383.5 + Y2 = 0 + 383.5

Reorder the terms:
-383.5 + 383.5 + 1.25Y + Y2 = 0 + 383.5

Combine like terms: -383.5 + 383.5 = 0.0
0.0 + 1.25Y + Y2 = 0 + 383.5
1.25Y + Y2 = 0 + 383.5

Combine like terms: 0 + 383.5 = 383.5
1.25Y + Y2 = 383.5

The Y term is 1.25Y.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25Y + 0.390625 + Y2 = 383.5 + 0.390625

Reorder the terms:
0.390625 + 1.25Y + Y2 = 383.5 + 0.390625

Combine like terms: 383.5 + 0.390625 = 383.890625
0.390625 + 1.25Y + Y2 = 383.890625

Factor a perfect square on the left side:
(Y + 0.625)(Y + 0.625) = 383.890625

Calculate the square root of the right side: 19.593126984

Break this problem into two subproblems by setting 
(Y + 0.625) equal to 19.593126984 and -19.593126984.


Subproblem 1
Y + 0.625 = 19.593126984

Simplifying
Y + 0.625 = 19.593126984

Reorder the terms:
0.625 + Y = 19.593126984

Solving
0.625 + Y = 19.593126984

Solving for variable 'Y'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + Y = 19.593126984 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + Y = 19.593126984 + -0.625
Y = 19.593126984 + -0.625

Combine like terms: 19.593126984 + -0.625 = 18.968126984
Y = 18.968126984

Simplifying
Y = 18.968126984


Subproblem 2
Y + 0.625 = -19.593126984

Simplifying
Y + 0.625 = -19.593126984

Reorder the terms:
0.625 + Y = -19.593126984

Solving
0.625 + Y = -19.593126984

Solving for variable 'Y'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + Y = -19.593126984 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + Y = -19.593126984 + -0.625
Y = -19.593126984 + -0.625

Combine like terms: -19.593126984 + -0.625 = -20.218126984
Y = -20.218126984

Simplifying
Y = -20.218126984


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {18.968126984, -20.218126984}
Solution
Y = {18.968126984, -20.218126984}

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