6y^2=10y+1

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Solution for 6y^2=10y+1 equation:


Simplifying
6y2 = 10y + 1

Reorder the terms:
6y2 = 1 + 10y

Solving
6y2 = 1 + 10y

Solving for variable 'y'.

Reorder the terms:
-1 + -10y + 6y2 = 1 + 10y + -1 + -10y

Reorder the terms:
-1 + -10y + 6y2 = 1 + -1 + 10y + -10y

Combine like terms: 1 + -1 = 0
-1 + -10y + 6y2 = 0 + 10y + -10y
-1 + -10y + 6y2 = 10y + -10y

Combine like terms: 10y + -10y = 0
-1 + -10y + 6y2 = 0

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

Divide each side by '6'.
-0.1666666667 + -1.666666667y + y2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.666666667y + 0.1666666667 + y2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.666666667y + y2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.666666667y + y2 = 0 + 0.1666666667
-1.666666667y + y2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.666666667y + y2 = 0.1666666667

The y term is -1.666666667y.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667y + 0.6944444447 + y2 = 0.1666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667y + y2 = 0.1666666667 + 0.6944444447

Combine like terms: 0.1666666667 + 0.6944444447 = 0.8611111114
0.6944444447 + -1.666666667y + y2 = 0.8611111114

Factor a perfect square on the left side:
(y + -0.8333333335)(y + -0.8333333335) = 0.8611111114

Calculate the square root of the right side: 0.927960727

Break this problem into two subproblems by setting 
(y + -0.8333333335) equal to 0.927960727 and -0.927960727.

Subproblem 1

y + -0.8333333335 = 0.927960727 Simplifying y + -0.8333333335 = 0.927960727 Reorder the terms: -0.8333333335 + y = 0.927960727 Solving -0.8333333335 + y = 0.927960727 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = 0.927960727 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = 0.927960727 + 0.8333333335 y = 0.927960727 + 0.8333333335 Combine like terms: 0.927960727 + 0.8333333335 = 1.7612940605 y = 1.7612940605 Simplifying y = 1.7612940605

Subproblem 2

y + -0.8333333335 = -0.927960727 Simplifying y + -0.8333333335 = -0.927960727 Reorder the terms: -0.8333333335 + y = -0.927960727 Solving -0.8333333335 + y = -0.927960727 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = -0.927960727 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = -0.927960727 + 0.8333333335 y = -0.927960727 + 0.8333333335 Combine like terms: -0.927960727 + 0.8333333335 = -0.0946273935 y = -0.0946273935 Simplifying y = -0.0946273935

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.7612940605, -0.0946273935}

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