6y^2+7y-45=0

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Solution for 6y^2+7y-45=0 equation:


Simplifying
6y2 + 7y + -45 = 0

Reorder the terms:
-45 + 7y + 6y2 = 0

Solving
-45 + 7y + 6y2 = 0

Solving for variable 'y'.

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

Divide each side by '6'.
-7.5 + 1.166666667y + y2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 1.166666667y + 7.5 + y2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 1.166666667y + y2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 1.166666667y + y2 = 0 + 7.5
1.166666667y + y2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
1.166666667y + y2 = 7.5

The y term is 1.166666667y.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667y + 0.3402777780 + y2 = 7.5 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667y + y2 = 7.5 + 0.3402777780

Combine like terms: 7.5 + 0.3402777780 = 7.840277778
0.3402777780 + 1.166666667y + y2 = 7.840277778

Factor a perfect square on the left side:
(y + 0.5833333335)(y + 0.5833333335) = 7.840277778

Calculate the square root of the right side: 2.800049603

Break this problem into two subproblems by setting 
(y + 0.5833333335) equal to 2.800049603 and -2.800049603.

Subproblem 1

y + 0.5833333335 = 2.800049603 Simplifying y + 0.5833333335 = 2.800049603 Reorder the terms: 0.5833333335 + y = 2.800049603 Solving 0.5833333335 + y = 2.800049603 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + y = 2.800049603 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + y = 2.800049603 + -0.5833333335 y = 2.800049603 + -0.5833333335 Combine like terms: 2.800049603 + -0.5833333335 = 2.2167162695 y = 2.2167162695 Simplifying y = 2.2167162695

Subproblem 2

y + 0.5833333335 = -2.800049603 Simplifying y + 0.5833333335 = -2.800049603 Reorder the terms: 0.5833333335 + y = -2.800049603 Solving 0.5833333335 + y = -2.800049603 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + y = -2.800049603 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + y = -2.800049603 + -0.5833333335 y = -2.800049603 + -0.5833333335 Combine like terms: -2.800049603 + -0.5833333335 = -3.3833829365 y = -3.3833829365 Simplifying y = -3.3833829365

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.2167162695, -3.3833829365}

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