3y^2+10y-51=0

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Solution for 3y^2+10y-51=0 equation:


Simplifying
3y2 + 10y + -51 = 0

Reorder the terms:
-51 + 10y + 3y2 = 0

Solving
-51 + 10y + 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'.
-17 + 3.333333333y + y2 = 0

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + 3.333333333y + 17 + y2 = 0 + 17

Reorder the terms:
-17 + 17 + 3.333333333y + y2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + 3.333333333y + y2 = 0 + 17
3.333333333y + y2 = 0 + 17

Combine like terms: 0 + 17 = 17
3.333333333y + y2 = 17

The y term is 3.333333333y.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333y + 2.777777779 + y2 = 17 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333y + y2 = 17 + 2.777777779

Combine like terms: 17 + 2.777777779 = 19.777777779
2.777777779 + 3.333333333y + y2 = 19.777777779

Factor a perfect square on the left side:
(y + 1.666666667)(y + 1.666666667) = 19.777777779

Calculate the square root of the right side: 4.447221355

Break this problem into two subproblems by setting 
(y + 1.666666667) equal to 4.447221355 and -4.447221355.

Subproblem 1

y + 1.666666667 = 4.447221355 Simplifying y + 1.666666667 = 4.447221355 Reorder the terms: 1.666666667 + y = 4.447221355 Solving 1.666666667 + y = 4.447221355 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = 4.447221355 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = 4.447221355 + -1.666666667 y = 4.447221355 + -1.666666667 Combine like terms: 4.447221355 + -1.666666667 = 2.780554688 y = 2.780554688 Simplifying y = 2.780554688

Subproblem 2

y + 1.666666667 = -4.447221355 Simplifying y + 1.666666667 = -4.447221355 Reorder the terms: 1.666666667 + y = -4.447221355 Solving 1.666666667 + y = -4.447221355 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = -4.447221355 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = -4.447221355 + -1.666666667 y = -4.447221355 + -1.666666667 Combine like terms: -4.447221355 + -1.666666667 = -6.113888022 y = -6.113888022 Simplifying y = -6.113888022

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.780554688, -6.113888022}

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