0=3y^2+14y-8

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Solution for 0=3y^2+14y-8 equation:


Simplifying
0 = 3y2 + 14y + -8

Reorder the terms:
0 = -8 + 14y + 3y2

Solving
0 = -8 + 14y + 3y2

Solving for variable 'y'.

Combine like terms: 0 + 8 = 8
8 + -14y + -3y2 = -8 + 14y + 3y2 + 8 + -14y + -3y2

Reorder the terms:
8 + -14y + -3y2 = -8 + 8 + 14y + -14y + 3y2 + -3y2

Combine like terms: -8 + 8 = 0
8 + -14y + -3y2 = 0 + 14y + -14y + 3y2 + -3y2
8 + -14y + -3y2 = 14y + -14y + 3y2 + -3y2

Combine like terms: 14y + -14y = 0
8 + -14y + -3y2 = 0 + 3y2 + -3y2
8 + -14y + -3y2 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
8 + -14y + -3y2 = 0

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

Divide each side by '-3'.
-2.666666667 + 4.666666667y + y2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + 4.666666667y + 2.666666667 + y2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + 4.666666667y + y2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + 4.666666667y + y2 = 0 + 2.666666667
4.666666667y + y2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
4.666666667y + y2 = 2.666666667

The y term is 4.666666667y.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667y + 5.444444448 + y2 = 2.666666667 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667y + y2 = 2.666666667 + 5.444444448

Combine like terms: 2.666666667 + 5.444444448 = 8.111111115
5.444444448 + 4.666666667y + y2 = 8.111111115

Factor a perfect square on the left side:
(y + 2.333333334)(y + 2.333333334) = 8.111111115

Calculate the square root of the right side: 2.848001249

Break this problem into two subproblems by setting 
(y + 2.333333334) equal to 2.848001249 and -2.848001249.

Subproblem 1

y + 2.333333334 = 2.848001249 Simplifying y + 2.333333334 = 2.848001249 Reorder the terms: 2.333333334 + y = 2.848001249 Solving 2.333333334 + y = 2.848001249 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + y = 2.848001249 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + y = 2.848001249 + -2.333333334 y = 2.848001249 + -2.333333334 Combine like terms: 2.848001249 + -2.333333334 = 0.514667915 y = 0.514667915 Simplifying y = 0.514667915

Subproblem 2

y + 2.333333334 = -2.848001249 Simplifying y + 2.333333334 = -2.848001249 Reorder the terms: 2.333333334 + y = -2.848001249 Solving 2.333333334 + y = -2.848001249 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + y = -2.848001249 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + y = -2.848001249 + -2.333333334 y = -2.848001249 + -2.333333334 Combine like terms: -2.848001249 + -2.333333334 = -5.181334583 y = -5.181334583 Simplifying y = -5.181334583

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.514667915, -5.181334583}

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