3y^2-y-3=0

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Solution for 3y^2-y-3=0 equation:


Simplifying
3y2 + -1y + -3 = 0

Reorder the terms:
-3 + -1y + 3y2 = 0

Solving
-3 + -1y + 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'.
-1 + -0.3333333333y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.3333333333y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.3333333333y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.3333333333y + y2 = 0 + 1
-0.3333333333y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.3333333333y + y2 = 1

The y term is -0.3333333333y.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333y + 0.02777777779 + y2 = 1 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333y + y2 = 1 + 0.02777777779

Combine like terms: 1 + 0.02777777779 = 1.02777777779
0.02777777779 + -0.3333333333y + y2 = 1.02777777779

Factor a perfect square on the left side:
(y + -0.1666666667)(y + -0.1666666667) = 1.02777777779

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(y + -0.1666666667) equal to 1.013793755 and -1.013793755.

Subproblem 1

y + -0.1666666667 = 1.013793755 Simplifying y + -0.1666666667 = 1.013793755 Reorder the terms: -0.1666666667 + y = 1.013793755 Solving -0.1666666667 + y = 1.013793755 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = 1.013793755 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = 1.013793755 + 0.1666666667 y = 1.013793755 + 0.1666666667 Combine like terms: 1.013793755 + 0.1666666667 = 1.1804604217 y = 1.1804604217 Simplifying y = 1.1804604217

Subproblem 2

y + -0.1666666667 = -1.013793755 Simplifying y + -0.1666666667 = -1.013793755 Reorder the terms: -0.1666666667 + y = -1.013793755 Solving -0.1666666667 + y = -1.013793755 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = -1.013793755 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = -1.013793755 + 0.1666666667 y = -1.013793755 + 0.1666666667 Combine like terms: -1.013793755 + 0.1666666667 = -0.8471270883 y = -0.8471270883 Simplifying y = -0.8471270883

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.1804604217, -0.8471270883}

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