3y(y-2)=11

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Solution for 3y(y-2)=11 equation:


Simplifying
3y(y + -2) = 11

Reorder the terms:
3y(-2 + y) = 11
(-2 * 3y + y * 3y) = 11
(-6y + 3y2) = 11

Solving
-6y + 3y2 = 11

Solving for variable 'y'.

Reorder the terms:
-11 + -6y + 3y2 = 11 + -11

Combine like terms: 11 + -11 = 0
-11 + -6y + 3y2 = 0

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

Divide each side by '3'.
-3.666666667 + -2y + y2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -2y + 3.666666667 + y2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -2y + y2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -2y + y2 = 0 + 3.666666667
-2y + y2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-2y + y2 = 3.666666667

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 = 3.666666667 + 1

Reorder the terms:
1 + -2y + y2 = 3.666666667 + 1

Combine like terms: 3.666666667 + 1 = 4.666666667
1 + -2y + y2 = 4.666666667

Factor a perfect square on the left side:
(y + -1)(y + -1) = 4.666666667

Calculate the square root of the right side: 2.1602469

Break this problem into two subproblems by setting 
(y + -1) equal to 2.1602469 and -2.1602469.

Subproblem 1

y + -1 = 2.1602469 Simplifying y + -1 = 2.1602469 Reorder the terms: -1 + y = 2.1602469 Solving -1 + y = 2.1602469 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 = 2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + y = 2.1602469 + 1 y = 2.1602469 + 1 Combine like terms: 2.1602469 + 1 = 3.1602469 y = 3.1602469 Simplifying y = 3.1602469

Subproblem 2

y + -1 = -2.1602469 Simplifying y + -1 = -2.1602469 Reorder the terms: -1 + y = -2.1602469 Solving -1 + y = -2.1602469 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 = -2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + y = -2.1602469 + 1 y = -2.1602469 + 1 Combine like terms: -2.1602469 + 1 = -1.1602469 y = -1.1602469 Simplifying y = -1.1602469

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.1602469, -1.1602469}

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