y^2-6y=5-6y^2

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Solution for y^2-6y=5-6y^2 equation:


Simplifying
y2 + -6y = 5 + -6y2

Reorder the terms:
-6y + y2 = 5 + -6y2

Solving
-6y + y2 = 5 + -6y2

Solving for variable 'y'.

Reorder the terms:
-5 + -6y + y2 + 6y2 = 5 + -6y2 + -5 + 6y2

Combine like terms: y2 + 6y2 = 7y2
-5 + -6y + 7y2 = 5 + -6y2 + -5 + 6y2

Reorder the terms:
-5 + -6y + 7y2 = 5 + -5 + -6y2 + 6y2

Combine like terms: 5 + -5 = 0
-5 + -6y + 7y2 = 0 + -6y2 + 6y2
-5 + -6y + 7y2 = -6y2 + 6y2

Combine like terms: -6y2 + 6y2 = 0
-5 + -6y + 7y2 = 0

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

Divide each side by '7'.
-0.7142857143 + -0.8571428571y + y2 = 0

Move the constant term to the right:

Add '0.7142857143' to each side of the equation.
-0.7142857143 + -0.8571428571y + 0.7142857143 + y2 = 0 + 0.7142857143

Reorder the terms:
-0.7142857143 + 0.7142857143 + -0.8571428571y + y2 = 0 + 0.7142857143

Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000
0.0000000000 + -0.8571428571y + y2 = 0 + 0.7142857143
-0.8571428571y + y2 = 0 + 0.7142857143

Combine like terms: 0 + 0.7142857143 = 0.7142857143
-0.8571428571y + y2 = 0.7142857143

The y term is -0.8571428571y.  Take half its coefficient (-0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
-0.8571428571y + 0.1836734694 + y2 = 0.7142857143 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571y + y2 = 0.7142857143 + 0.1836734694

Combine like terms: 0.7142857143 + 0.1836734694 = 0.8979591837
0.1836734694 + -0.8571428571y + y2 = 0.8979591837

Factor a perfect square on the left side:
(y + -0.4285714286)(y + -0.4285714286) = 0.8979591837

Calculate the square root of the right side: 0.947607083

Break this problem into two subproblems by setting 
(y + -0.4285714286) equal to 0.947607083 and -0.947607083.

Subproblem 1

y + -0.4285714286 = 0.947607083 Simplifying y + -0.4285714286 = 0.947607083 Reorder the terms: -0.4285714286 + y = 0.947607083 Solving -0.4285714286 + y = 0.947607083 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + y = 0.947607083 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + y = 0.947607083 + 0.4285714286 y = 0.947607083 + 0.4285714286 Combine like terms: 0.947607083 + 0.4285714286 = 1.3761785116 y = 1.3761785116 Simplifying y = 1.3761785116

Subproblem 2

y + -0.4285714286 = -0.947607083 Simplifying y + -0.4285714286 = -0.947607083 Reorder the terms: -0.4285714286 + y = -0.947607083 Solving -0.4285714286 + y = -0.947607083 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + y = -0.947607083 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + y = -0.947607083 + 0.4285714286 y = -0.947607083 + 0.4285714286 Combine like terms: -0.947607083 + 0.4285714286 = -0.5190356544 y = -0.5190356544 Simplifying y = -0.5190356544

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.3761785116, -0.5190356544}

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