3y^2+2y-45=0

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Solution for 3y^2+2y-45=0 equation:


Simplifying
3y2 + 2y + -45 = 0

Reorder the terms:
-45 + 2y + 3y2 = 0

Solving
-45 + 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'.
-15 + 0.6666666667y + y2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 0.6666666667y + 15 + y2 = 0 + 15

Reorder the terms:
-15 + 15 + 0.6666666667y + y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 0.6666666667y + y2 = 0 + 15
0.6666666667y + y2 = 0 + 15

Combine like terms: 0 + 15 = 15
0.6666666667y + y2 = 15

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 = 15 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667y + y2 = 15 + 0.1111111112

Combine like terms: 15 + 0.1111111112 = 15.1111111112
0.1111111112 + 0.6666666667y + y2 = 15.1111111112

Factor a perfect square on the left side:
(y + 0.3333333334)(y + 0.3333333334) = 15.1111111112

Calculate the square root of the right side: 3.887301263

Break this problem into two subproblems by setting 
(y + 0.3333333334) equal to 3.887301263 and -3.887301263.

Subproblem 1

y + 0.3333333334 = 3.887301263 Simplifying y + 0.3333333334 = 3.887301263 Reorder the terms: 0.3333333334 + y = 3.887301263 Solving 0.3333333334 + y = 3.887301263 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 = 3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = 3.887301263 + -0.3333333334 y = 3.887301263 + -0.3333333334 Combine like terms: 3.887301263 + -0.3333333334 = 3.5539679296 y = 3.5539679296 Simplifying y = 3.5539679296

Subproblem 2

y + 0.3333333334 = -3.887301263 Simplifying y + 0.3333333334 = -3.887301263 Reorder the terms: 0.3333333334 + y = -3.887301263 Solving 0.3333333334 + y = -3.887301263 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 = -3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = -3.887301263 + -0.3333333334 y = -3.887301263 + -0.3333333334 Combine like terms: -3.887301263 + -0.3333333334 = -4.2206345964 y = -4.2206345964 Simplifying y = -4.2206345964

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.5539679296, -4.2206345964}

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