5Y^2-Y-12=0

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Solution for 5Y^2-Y-12=0 equation:


Simplifying
5Y2 + -1Y + -12 = 0

Reorder the terms:
-12 + -1Y + 5Y2 = 0

Solving
-12 + -1Y + 5Y2 = 0

Solving for variable 'Y'.

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

Divide each side by '5'.
-2.4 + -0.2Y + Y2 = 0

Move the constant term to the right:

Add '2.4' to each side of the equation.
-2.4 + -0.2Y + 2.4 + Y2 = 0 + 2.4

Reorder the terms:
-2.4 + 2.4 + -0.2Y + Y2 = 0 + 2.4

Combine like terms: -2.4 + 2.4 = 0.0
0.0 + -0.2Y + Y2 = 0 + 2.4
-0.2Y + Y2 = 0 + 2.4

Combine like terms: 0 + 2.4 = 2.4
-0.2Y + Y2 = 2.4

The Y term is -0.2Y.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2Y + 0.01 + Y2 = 2.4 + 0.01

Reorder the terms:
0.01 + -0.2Y + Y2 = 2.4 + 0.01

Combine like terms: 2.4 + 0.01 = 2.41
0.01 + -0.2Y + Y2 = 2.41

Factor a perfect square on the left side:
(Y + -0.1)(Y + -0.1) = 2.41

Calculate the square root of the right side: 1.55241747

Break this problem into two subproblems by setting 
(Y + -0.1) equal to 1.55241747 and -1.55241747.

Subproblem 1

Y + -0.1 = 1.55241747 Simplifying Y + -0.1 = 1.55241747 Reorder the terms: -0.1 + Y = 1.55241747 Solving -0.1 + Y = 1.55241747 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + Y = 1.55241747 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + Y = 1.55241747 + 0.1 Y = 1.55241747 + 0.1 Combine like terms: 1.55241747 + 0.1 = 1.65241747 Y = 1.65241747 Simplifying Y = 1.65241747

Subproblem 2

Y + -0.1 = -1.55241747 Simplifying Y + -0.1 = -1.55241747 Reorder the terms: -0.1 + Y = -1.55241747 Solving -0.1 + Y = -1.55241747 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + Y = -1.55241747 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + Y = -1.55241747 + 0.1 Y = -1.55241747 + 0.1 Combine like terms: -1.55241747 + 0.1 = -1.45241747 Y = -1.45241747 Simplifying Y = -1.45241747

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {1.65241747, -1.45241747}

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