-6y^2+16y=-45

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Solution for -6y^2+16y=-45 equation:


Simplifying
-6y2 + 16y = -45

Reorder the terms:
16y + -6y2 = -45

Solving
16y + -6y2 = -45

Solving for variable 'y'.

Reorder the terms:
45 + 16y + -6y2 = -45 + 45

Combine like terms: -45 + 45 = 0
45 + 16y + -6y2 = 0

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

Divide each side by '-6'.
-7.5 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -2.666666667y + 7.5 + y2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -2.666666667y + y2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -2.666666667y + y2 = 0 + 7.5
-2.666666667y + y2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-2.666666667y + y2 = 7.5

The y term is -2.666666667y.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667y + 1.777777780 + y2 = 7.5 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = 7.5 + 1.777777780

Combine like terms: 7.5 + 1.777777780 = 9.27777778
1.777777780 + -2.666666667y + y2 = 9.27777778

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 9.27777778

Calculate the square root of the right side: 3.045944481

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 3.045944481 and -3.045944481.

Subproblem 1

y + -1.333333334 = 3.045944481 Simplifying y + -1.333333334 = 3.045944481 Reorder the terms: -1.333333334 + y = 3.045944481 Solving -1.333333334 + y = 3.045944481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 3.045944481 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 3.045944481 + 1.333333334 y = 3.045944481 + 1.333333334 Combine like terms: 3.045944481 + 1.333333334 = 4.379277815 y = 4.379277815 Simplifying y = 4.379277815

Subproblem 2

y + -1.333333334 = -3.045944481 Simplifying y + -1.333333334 = -3.045944481 Reorder the terms: -1.333333334 + y = -3.045944481 Solving -1.333333334 + y = -3.045944481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -3.045944481 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -3.045944481 + 1.333333334 y = -3.045944481 + 1.333333334 Combine like terms: -3.045944481 + 1.333333334 = -1.712611147 y = -1.712611147 Simplifying y = -1.712611147

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.379277815, -1.712611147}

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