(3y+5)(3y+5)+(y-4)(y-4)=6y(6y)

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Solution for (3y+5)(3y+5)+(y-4)(y-4)=6y(6y) equation:


Simplifying
(3y + 5)(3y + 5) + (y + -4)(y + -4) = 6y(6y)

Reorder the terms:
(5 + 3y)(3y + 5) + (y + -4)(y + -4) = 6y(6y)

Reorder the terms:
(5 + 3y)(5 + 3y) + (y + -4)(y + -4) = 6y(6y)

Multiply (5 + 3y) * (5 + 3y)
(5(5 + 3y) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y)
((5 * 5 + 3y * 5) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y)
((25 + 15y) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y)
(25 + 15y + (5 * 3y + 3y * 3y)) + (y + -4)(y + -4) = 6y(6y)
(25 + 15y + (15y + 9y2)) + (y + -4)(y + -4) = 6y(6y)

Combine like terms: 15y + 15y = 30y
(25 + 30y + 9y2) + (y + -4)(y + -4) = 6y(6y)

Reorder the terms:
25 + 30y + 9y2 + (-4 + y)(y + -4) = 6y(6y)

Reorder the terms:
25 + 30y + 9y2 + (-4 + y)(-4 + y) = 6y(6y)

Multiply (-4 + y) * (-4 + y)
25 + 30y + 9y2 + (-4(-4 + y) + y(-4 + y)) = 6y(6y)
25 + 30y + 9y2 + ((-4 * -4 + y * -4) + y(-4 + y)) = 6y(6y)
25 + 30y + 9y2 + ((16 + -4y) + y(-4 + y)) = 6y(6y)
25 + 30y + 9y2 + (16 + -4y + (-4 * y + y * y)) = 6y(6y)
25 + 30y + 9y2 + (16 + -4y + (-4y + y2)) = 6y(6y)

Combine like terms: -4y + -4y = -8y
25 + 30y + 9y2 + (16 + -8y + y2) = 6y(6y)

Reorder the terms:
25 + 16 + 30y + -8y + 9y2 + y2 = 6y(6y)

Combine like terms: 25 + 16 = 41
41 + 30y + -8y + 9y2 + y2 = 6y(6y)

Combine like terms: 30y + -8y = 22y
41 + 22y + 9y2 + y2 = 6y(6y)

Combine like terms: 9y2 + y2 = 10y2
41 + 22y + 10y2 = 6y(6y)

Remove parenthesis around (6y)
41 + 22y + 10y2 = 6y * 6y

Reorder the terms for easier multiplication:
41 + 22y + 10y2 = 6 * 6y * y

Multiply 6 * 6
41 + 22y + 10y2 = 36y * y

Multiply y * y
41 + 22y + 10y2 = 36y2

Solving
41 + 22y + 10y2 = 36y2

Solving for variable 'y'.

Combine like terms: 10y2 + -36y2 = -26y2
41 + 22y + -26y2 = 36y2 + -36y2

Combine like terms: 36y2 + -36y2 = 0
41 + 22y + -26y2 = 0

Begin completing the square.  Divide all terms by
-26 the coefficient of the squared term: 

Divide each side by '-26'.
-1.576923077 + -0.8461538462y + y2 = 0

Move the constant term to the right:

Add '1.576923077' to each side of the equation.
-1.576923077 + -0.8461538462y + 1.576923077 + y2 = 0 + 1.576923077

Reorder the terms:
-1.576923077 + 1.576923077 + -0.8461538462y + y2 = 0 + 1.576923077

Combine like terms: -1.576923077 + 1.576923077 = 0.000000000
0.000000000 + -0.8461538462y + y2 = 0 + 1.576923077
-0.8461538462y + y2 = 0 + 1.576923077

Combine like terms: 0 + 1.576923077 = 1.576923077
-0.8461538462y + y2 = 1.576923077

The y term is -0.8461538462y.  Take half its coefficient (-0.4230769231).
Square it (0.1789940829) and add it to both sides.

Add '0.1789940829' to each side of the equation.
-0.8461538462y + 0.1789940829 + y2 = 1.576923077 + 0.1789940829

Reorder the terms:
0.1789940829 + -0.8461538462y + y2 = 1.576923077 + 0.1789940829

Combine like terms: 1.576923077 + 0.1789940829 = 1.7559171599
0.1789940829 + -0.8461538462y + y2 = 1.7559171599

Factor a perfect square on the left side:
(y + -0.4230769231)(y + -0.4230769231) = 1.7559171599

Calculate the square root of the right side: 1.325110244

Break this problem into two subproblems by setting 
(y + -0.4230769231) equal to 1.325110244 and -1.325110244.

Subproblem 1

y + -0.4230769231 = 1.325110244 Simplifying y + -0.4230769231 = 1.325110244 Reorder the terms: -0.4230769231 + y = 1.325110244 Solving -0.4230769231 + y = 1.325110244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + y = 1.325110244 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + y = 1.325110244 + 0.4230769231 y = 1.325110244 + 0.4230769231 Combine like terms: 1.325110244 + 0.4230769231 = 1.7481871671 y = 1.7481871671 Simplifying y = 1.7481871671

Subproblem 2

y + -0.4230769231 = -1.325110244 Simplifying y + -0.4230769231 = -1.325110244 Reorder the terms: -0.4230769231 + y = -1.325110244 Solving -0.4230769231 + y = -1.325110244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + y = -1.325110244 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + y = -1.325110244 + 0.4230769231 y = -1.325110244 + 0.4230769231 Combine like terms: -1.325110244 + 0.4230769231 = -0.9020333209 y = -0.9020333209 Simplifying y = -0.9020333209

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.7481871671, -0.9020333209}

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