6y^2-54y-42=0

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Solution for 6y^2-54y-42=0 equation: 


Simplifying
6y2 + -54y + -42 = 0

Reorder the terms:
-42 + -54y + 6y2 = 0

Solving
-42 + -54y + 6y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-7 + -9y + y2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-7 + -9y + y2)' equal to zero and attempt to solve:

Simplifying
-7 + -9y + y2 = 0

Solving
-7 + -9y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -9y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + -9y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -9y + y2 = 0 + 7
-9y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
-9y + y2 = 7

The y term is -9y.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9y + 20.25 + y2 = 7 + 20.25

Reorder the terms:
20.25 + -9y + y2 = 7 + 20.25

Combine like terms: 7 + 20.25 = 27.25
20.25 + -9y + y2 = 27.25

Factor a perfect square on the left side:
(y + -4.5)(y + -4.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(y + -4.5) equal to 5.220153254 and -5.220153254.


Subproblem 1
y + -4.5 = 5.220153254

Simplifying
y + -4.5 = 5.220153254

Reorder the terms:
-4.5 + y = 5.220153254

Solving
-4.5 + y = 5.220153254

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '4.5' to each side of the equation.
-4.5 + 4.5 + y = 5.220153254 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + y = 5.220153254 + 4.5
y = 5.220153254 + 4.5

Combine like terms: 5.220153254 + 4.5 = 9.720153254
y = 9.720153254

Simplifying
y = 9.720153254


Subproblem 2
y + -4.5 = -5.220153254

Simplifying
y + -4.5 = -5.220153254

Reorder the terms:
-4.5 + y = -5.220153254

Solving
-4.5 + y = -5.220153254

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '4.5' to each side of the equation.
-4.5 + 4.5 + y = -5.220153254 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + y = -5.220153254 + 4.5
y = -5.220153254 + 4.5

Combine like terms: -5.220153254 + 4.5 = -0.720153254
y = -0.720153254

Simplifying
y = -0.720153254


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {9.720153254, -0.720153254}
Solution
y = {9.720153254, -0.720153254}

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