13y^2-32y-12=0

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Solution for 13y^2-32y-12=0 equation:


Simplifying
13y2 + -32y + -12 = 0

Reorder the terms:
-12 + -32y + 13y2 = 0

Solving
-12 + -32y + 13y2 = 0

Solving for variable 'y'.

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

Divide each side by '13'.
-0.9230769231 + -2.461538462y + y2 = 0

Move the constant term to the right:

Add '0.9230769231' to each side of the equation.
-0.9230769231 + -2.461538462y + 0.9230769231 + y2 = 0 + 0.9230769231

Reorder the terms:
-0.9230769231 + 0.9230769231 + -2.461538462y + y2 = 0 + 0.9230769231

Combine like terms: -0.9230769231 + 0.9230769231 = 0.0000000000
0.0000000000 + -2.461538462y + y2 = 0 + 0.9230769231
-2.461538462y + y2 = 0 + 0.9230769231

Combine like terms: 0 + 0.9230769231 = 0.9230769231
-2.461538462y + y2 = 0.9230769231

The y term is -2.461538462y.  Take half its coefficient (-1.230769231).
Square it (1.514792900) and add it to both sides.

Add '1.514792900' to each side of the equation.
-2.461538462y + 1.514792900 + y2 = 0.9230769231 + 1.514792900

Reorder the terms:
1.514792900 + -2.461538462y + y2 = 0.9230769231 + 1.514792900

Combine like terms: 0.9230769231 + 1.514792900 = 2.4378698231
1.514792900 + -2.461538462y + y2 = 2.4378698231

Factor a perfect square on the left side:
(y + -1.230769231)(y + -1.230769231) = 2.4378698231

Calculate the square root of the right side: 1.561367933

Break this problem into two subproblems by setting 
(y + -1.230769231) equal to 1.561367933 and -1.561367933.

Subproblem 1

y + -1.230769231 = 1.561367933 Simplifying y + -1.230769231 = 1.561367933 Reorder the terms: -1.230769231 + y = 1.561367933 Solving -1.230769231 + y = 1.561367933 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.230769231' to each side of the equation. -1.230769231 + 1.230769231 + y = 1.561367933 + 1.230769231 Combine like terms: -1.230769231 + 1.230769231 = 0.000000000 0.000000000 + y = 1.561367933 + 1.230769231 y = 1.561367933 + 1.230769231 Combine like terms: 1.561367933 + 1.230769231 = 2.792137164 y = 2.792137164 Simplifying y = 2.792137164

Subproblem 2

y + -1.230769231 = -1.561367933 Simplifying y + -1.230769231 = -1.561367933 Reorder the terms: -1.230769231 + y = -1.561367933 Solving -1.230769231 + y = -1.561367933 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.230769231' to each side of the equation. -1.230769231 + 1.230769231 + y = -1.561367933 + 1.230769231 Combine like terms: -1.230769231 + 1.230769231 = 0.000000000 0.000000000 + y = -1.561367933 + 1.230769231 y = -1.561367933 + 1.230769231 Combine like terms: -1.561367933 + 1.230769231 = -0.330598702 y = -0.330598702 Simplifying y = -0.330598702

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.792137164, -0.330598702}

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