3y^2-y-21=0

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Solution for 3y^2-y-21=0 equation:


Simplifying
3y2 + -1y + -21 = 0

Reorder the terms:
-21 + -1y + 3y2 = 0

Solving
-21 + -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'.
-7 + -0.3333333333y + y2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -0.3333333333y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + -0.3333333333y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -0.3333333333y + y2 = 0 + 7
-0.3333333333y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
-0.3333333333y + y2 = 7

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 = 7 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333y + y2 = 7 + 0.02777777779

Combine like terms: 7 + 0.02777777779 = 7.02777777779
0.02777777779 + -0.3333333333y + y2 = 7.02777777779

Factor a perfect square on the left side:
(y + -0.1666666667)(y + -0.1666666667) = 7.02777777779

Calculate the square root of the right side: 2.65099562

Break this problem into two subproblems by setting 
(y + -0.1666666667) equal to 2.65099562 and -2.65099562.

Subproblem 1

y + -0.1666666667 = 2.65099562 Simplifying y + -0.1666666667 = 2.65099562 Reorder the terms: -0.1666666667 + y = 2.65099562 Solving -0.1666666667 + y = 2.65099562 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 = 2.65099562 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = 2.65099562 + 0.1666666667 y = 2.65099562 + 0.1666666667 Combine like terms: 2.65099562 + 0.1666666667 = 2.8176622867 y = 2.8176622867 Simplifying y = 2.8176622867

Subproblem 2

y + -0.1666666667 = -2.65099562 Simplifying y + -0.1666666667 = -2.65099562 Reorder the terms: -0.1666666667 + y = -2.65099562 Solving -0.1666666667 + y = -2.65099562 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 = -2.65099562 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = -2.65099562 + 0.1666666667 y = -2.65099562 + 0.1666666667 Combine like terms: -2.65099562 + 0.1666666667 = -2.4843289533 y = -2.4843289533 Simplifying y = -2.4843289533

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.8176622867, -2.4843289533}

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