6y^2+y=6

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Solution for 6y^2+y=6 equation:


Simplifying
6y2 + y = 6

Reorder the terms:
y + 6y2 = 6

Solving
y + 6y2 = 6

Solving for variable 'y'.

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

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

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

Divide each side by '6'.
-1 + 0.1666666667y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.1666666667y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.1666666667y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.1666666667y + y2 = 0 + 1
0.1666666667y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.1666666667y + y2 = 1

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667y + 0.25 + y2 = 1 + 0.25

Reorder the terms:
0.25 + 0.1666666667y + y2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + 0.1666666667y + y2 = 1.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(y + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

y + 0.5 = 1.118033989 Simplifying y + 0.5 = 1.118033989 Reorder the terms: 0.5 + y = 1.118033989 Solving 0.5 + y = 1.118033989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.118033989 + -0.5 y = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 y = 0.618033989 Simplifying y = 0.618033989

Subproblem 2

y + 0.5 = -1.118033989 Simplifying y + 0.5 = -1.118033989 Reorder the terms: 0.5 + y = -1.118033989 Solving 0.5 + y = -1.118033989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.118033989 + -0.5 y = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 y = -1.618033989 Simplifying y = -1.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.618033989, -1.618033989}

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