3y^2-18y+19=0

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Solution for 3y^2-18y+19=0 equation:


Simplifying
3y2 + -18y + 19 = 0

Reorder the terms:
19 + -18y + 3y2 = 0

Solving
19 + -18y + 3y2 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
6.333333333 + -6y + y2 = 0

Move the constant term to the right:

Add '-6.333333333' to each side of the equation.
6.333333333 + -6y + -6.333333333 + y2 = 0 + -6.333333333

Reorder the terms:
6.333333333 + -6.333333333 + -6y + y2 = 0 + -6.333333333

Combine like terms: 6.333333333 + -6.333333333 = 0.000000000
0.000000000 + -6y + y2 = 0 + -6.333333333
-6y + y2 = 0 + -6.333333333

Combine like terms: 0 + -6.333333333 = -6.333333333
-6y + y2 = -6.333333333

The y term is -6y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6y + 9 + y2 = -6.333333333 + 9

Reorder the terms:
9 + -6y + y2 = -6.333333333 + 9

Combine like terms: -6.333333333 + 9 = 2.666666667
9 + -6y + y2 = 2.666666667

Factor a perfect square on the left side:
(y + -3)(y + -3) = 2.666666667

Calculate the square root of the right side: 1.632993162

Break this problem into two subproblems by setting 
(y + -3) equal to 1.632993162 and -1.632993162.

Subproblem 1

y + -3 = 1.632993162 Simplifying y + -3 = 1.632993162 Reorder the terms: -3 + y = 1.632993162 Solving -3 + y = 1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = 1.632993162 + 3 Combine like terms: -3 + 3 = 0 0 + y = 1.632993162 + 3 y = 1.632993162 + 3 Combine like terms: 1.632993162 + 3 = 4.632993162 y = 4.632993162 Simplifying y = 4.632993162

Subproblem 2

y + -3 = -1.632993162 Simplifying y + -3 = -1.632993162 Reorder the terms: -3 + y = -1.632993162 Solving -3 + y = -1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = -1.632993162 + 3 Combine like terms: -3 + 3 = 0 0 + y = -1.632993162 + 3 y = -1.632993162 + 3 Combine like terms: -1.632993162 + 3 = 1.367006838 y = 1.367006838 Simplifying y = 1.367006838

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.632993162, 1.367006838}

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