(5Y^2-6Y+8)-(-3Y^2-7Y+8)+(7Y^2+6Y-9)=

Solution for (5Y^2-6Y+8)-(-3Y^2-7Y+8)+(7Y^2+6Y-9)= equation:

Simplifying
(5Y2 + -6Y + 8) + -1(-3Y2 + -7Y + 8) + (7Y2 + 6Y + -9) = 0

Reorder the terms:
(8 + -6Y + 5Y2) + -1(-3Y2 + -7Y + 8) + (7Y2 + 6Y + -9) = 0

Remove parenthesis around (8 + -6Y + 5Y2)
8 + -6Y + 5Y2 + -1(-3Y2 + -7Y + 8) + (7Y2 + 6Y + -9) = 0

Reorder the terms:
8 + -6Y + 5Y2 + -1(8 + -7Y + -3Y2) + (7Y2 + 6Y + -9) = 0
8 + -6Y + 5Y2 + (8 * -1 + -7Y * -1 + -3Y2 * -1) + (7Y2 + 6Y + -9) = 0
8 + -6Y + 5Y2 + (-8 + 7Y + 3Y2) + (7Y2 + 6Y + -9) = 0

Reorder the terms:
8 + -6Y + 5Y2 + -8 + 7Y + 3Y2 + (-9 + 6Y + 7Y2) = 0

Remove parenthesis around (-9 + 6Y + 7Y2)
8 + -6Y + 5Y2 + -8 + 7Y + 3Y2 + -9 + 6Y + 7Y2 = 0

Reorder the terms:
8 + -8 + -9 + -6Y + 7Y + 6Y + 5Y2 + 3Y2 + 7Y2 = 0

Combine like terms: 8 + -8 = 0
0 + -9 + -6Y + 7Y + 6Y + 5Y2 + 3Y2 + 7Y2 = 0
-9 + -6Y + 7Y + 6Y + 5Y2 + 3Y2 + 7Y2 = 0

Combine like terms: -6Y + 7Y = 1Y
-9 + 1Y + 6Y + 5Y2 + 3Y2 + 7Y2 = 0

Combine like terms: 1Y + 6Y = 7Y
-9 + 7Y + 5Y2 + 3Y2 + 7Y2 = 0

Combine like terms: 5Y2 + 3Y2 = 8Y2
-9 + 7Y + 8Y2 + 7Y2 = 0

Combine like terms: 8Y2 + 7Y2 = 15Y2
-9 + 7Y + 15Y2 = 0

Solving
-9 + 7Y + 15Y2 = 0

Solving for variable 'Y'.

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

Divide each side by '15'.
-0.6 + 0.4666666667Y + Y2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + 0.4666666667Y + 0.6 + Y2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + 0.4666666667Y + Y2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + 0.4666666667Y + Y2 = 0 + 0.6
0.4666666667Y + Y2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
0.4666666667Y + Y2 = 0.6

The Y term is 0.4666666667Y.  Take half its coefficient (0.2333333334).
Square it (0.05444444448) and add it to both sides.

Add '0.05444444448' to each side of the equation.
0.4666666667Y + 0.05444444448 + Y2 = 0.6 + 0.05444444448

Reorder the terms:
0.05444444448 + 0.4666666667Y + Y2 = 0.6 + 0.05444444448

Combine like terms: 0.6 + 0.05444444448 = 0.65444444448
0.05444444448 + 0.4666666667Y + Y2 = 0.65444444448

Factor a perfect square on the left side:
(Y + 0.2333333334)(Y + 0.2333333334) = 0.65444444448

Calculate the square root of the right side: 0.808977407

Break this problem into two subproblems by setting 
(Y + 0.2333333334) equal to 0.808977407 and -0.808977407.

Subproblem 1
Y + 0.2333333334 = 0.808977407

Simplifying
Y + 0.2333333334 = 0.808977407

Reorder the terms:
0.2333333334 + Y = 0.808977407

Solving
0.2333333334 + Y = 0.808977407

Solving for variable 'Y'.

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

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + Y = 0.808977407 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + Y = 0.808977407 + -0.2333333334
Y = 0.808977407 + -0.2333333334

Combine like terms: 0.808977407 + -0.2333333334 = 0.5756440736
Y = 0.5756440736

Simplifying
Y = 0.5756440736

Subproblem 2
Y + 0.2333333334 = -0.808977407

Simplifying
Y + 0.2333333334 = -0.808977407

Reorder the terms:
0.2333333334 + Y = -0.808977407

Solving
0.2333333334 + Y = -0.808977407

Solving for variable 'Y'.

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

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + Y = -0.808977407 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + Y = -0.808977407 + -0.2333333334
Y = -0.808977407 + -0.2333333334

Combine like terms: -0.808977407 + -0.2333333334 = -1.0423107404
Y = -1.0423107404

Simplifying
Y = -1.0423107404

Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {0.5756440736, -1.0423107404}