3y^2+2y-75=0

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Solution for 3y^2+2y-75=0 equation:


Simplifying
3y2 + 2y + -75 = 0

Reorder the terms:
-75 + 2y + 3y2 = 0

Solving
-75 + 2y + 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'.
-25 + 0.6666666667y + y2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 0.6666666667y + 25 + y2 = 0 + 25

Reorder the terms:
-25 + 25 + 0.6666666667y + y2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 0.6666666667y + y2 = 0 + 25
0.6666666667y + y2 = 0 + 25

Combine like terms: 0 + 25 = 25
0.6666666667y + y2 = 25

The y term is 0.6666666667y.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667y + 0.1111111112 + y2 = 25 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667y + y2 = 25 + 0.1111111112

Combine like terms: 25 + 0.1111111112 = 25.1111111112
0.1111111112 + 0.6666666667y + y2 = 25.1111111112

Factor a perfect square on the left side:
(y + 0.3333333334)(y + 0.3333333334) = 25.1111111112

Calculate the square root of the right side: 5.011098793

Break this problem into two subproblems by setting 
(y + 0.3333333334) equal to 5.011098793 and -5.011098793.

Subproblem 1

y + 0.3333333334 = 5.011098793 Simplifying y + 0.3333333334 = 5.011098793 Reorder the terms: 0.3333333334 + y = 5.011098793 Solving 0.3333333334 + y = 5.011098793 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + y = 5.011098793 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = 5.011098793 + -0.3333333334 y = 5.011098793 + -0.3333333334 Combine like terms: 5.011098793 + -0.3333333334 = 4.6777654596 y = 4.6777654596 Simplifying y = 4.6777654596

Subproblem 2

y + 0.3333333334 = -5.011098793 Simplifying y + 0.3333333334 = -5.011098793 Reorder the terms: 0.3333333334 + y = -5.011098793 Solving 0.3333333334 + y = -5.011098793 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + y = -5.011098793 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = -5.011098793 + -0.3333333334 y = -5.011098793 + -0.3333333334 Combine like terms: -5.011098793 + -0.3333333334 = -5.3444321264 y = -5.3444321264 Simplifying y = -5.3444321264

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.6777654596, -5.3444321264}

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