Y^2-18Y+6=0

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Solution for Y^2-18Y+6=0 equation:


Simplifying
Y2 + -18Y + 6 = 0

Reorder the terms:
6 + -18Y + Y2 = 0

Solving
6 + -18Y + Y2 = 0

Solving for variable 'Y'.

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -18Y + -6 + Y2 = 0 + -6

Reorder the terms:
6 + -6 + -18Y + Y2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -18Y + Y2 = 0 + -6
-18Y + Y2 = 0 + -6

Combine like terms: 0 + -6 = -6
-18Y + Y2 = -6

The Y term is -18Y.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18Y + 81 + Y2 = -6 + 81

Reorder the terms:
81 + -18Y + Y2 = -6 + 81

Combine like terms: -6 + 81 = 75
81 + -18Y + Y2 = 75

Factor a perfect square on the left side:
(Y + -9)(Y + -9) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(Y + -9) equal to 8.660254038 and -8.660254038.

Subproblem 1

Y + -9 = 8.660254038 Simplifying Y + -9 = 8.660254038 Reorder the terms: -9 + Y = 8.660254038 Solving -9 + Y = 8.660254038 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + Y = 8.660254038 + 9 Combine like terms: -9 + 9 = 0 0 + Y = 8.660254038 + 9 Y = 8.660254038 + 9 Combine like terms: 8.660254038 + 9 = 17.660254038 Y = 17.660254038 Simplifying Y = 17.660254038

Subproblem 2

Y + -9 = -8.660254038 Simplifying Y + -9 = -8.660254038 Reorder the terms: -9 + Y = -8.660254038 Solving -9 + Y = -8.660254038 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + Y = -8.660254038 + 9 Combine like terms: -9 + 9 = 0 0 + Y = -8.660254038 + 9 Y = -8.660254038 + 9 Combine like terms: -8.660254038 + 9 = 0.339745962 Y = 0.339745962 Simplifying Y = 0.339745962

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {17.660254038, 0.339745962}

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