3y^2=-5y+6

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Solution for 3y^2=-5y+6 equation:


Simplifying
3y2 = -5y + 6

Reorder the terms:
3y2 = 6 + -5y

Solving
3y2 = 6 + -5y

Solving for variable 'y'.

Reorder the terms:
-6 + 5y + 3y2 = 6 + -5y + -6 + 5y

Reorder the terms:
-6 + 5y + 3y2 = 6 + -6 + -5y + 5y

Combine like terms: 6 + -6 = 0
-6 + 5y + 3y2 = 0 + -5y + 5y
-6 + 5y + 3y2 = -5y + 5y

Combine like terms: -5y + 5y = 0
-6 + 5y + 3y2 = 0

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

Divide each side by '3'.
-2 + 1.666666667y + y2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 1.666666667y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + 1.666666667y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 1.666666667y + y2 = 0 + 2
1.666666667y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
1.666666667y + y2 = 2

The y term is 1.666666667y.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667y + 0.6944444447 + y2 = 2 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667y + y2 = 2 + 0.6944444447

Combine like terms: 2 + 0.6944444447 = 2.6944444447
0.6944444447 + 1.666666667y + y2 = 2.6944444447

Factor a perfect square on the left side:
(y + 0.8333333335)(y + 0.8333333335) = 2.6944444447

Calculate the square root of the right side: 1.6414763

Break this problem into two subproblems by setting 
(y + 0.8333333335) equal to 1.6414763 and -1.6414763.

Subproblem 1

y + 0.8333333335 = 1.6414763 Simplifying y + 0.8333333335 = 1.6414763 Reorder the terms: 0.8333333335 + y = 1.6414763 Solving 0.8333333335 + y = 1.6414763 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = 1.6414763 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = 1.6414763 + -0.8333333335 y = 1.6414763 + -0.8333333335 Combine like terms: 1.6414763 + -0.8333333335 = 0.8081429665 y = 0.8081429665 Simplifying y = 0.8081429665

Subproblem 2

y + 0.8333333335 = -1.6414763 Simplifying y + 0.8333333335 = -1.6414763 Reorder the terms: 0.8333333335 + y = -1.6414763 Solving 0.8333333335 + y = -1.6414763 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = -1.6414763 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = -1.6414763 + -0.8333333335 y = -1.6414763 + -0.8333333335 Combine like terms: -1.6414763 + -0.8333333335 = -2.4748096335 y = -2.4748096335 Simplifying y = -2.4748096335

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.8081429665, -2.4748096335}

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