Y^2-6Y=-3

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Solution for Y^2-6Y=-3 equation:


Simplifying
Y2 + -6Y = -3

Reorder the terms:
-6Y + Y2 = -3

Solving
-6Y + Y2 = -3

Solving for variable 'Y'.

Reorder the terms:
3 + -6Y + Y2 = -3 + 3

Combine like terms: -3 + 3 = 0
3 + -6Y + Y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -6Y + -3 + Y2 = 0 + -3

Reorder the terms:
3 + -3 + -6Y + Y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -6Y + Y2 = 0 + -3
-6Y + Y2 = 0 + -3

Combine like terms: 0 + -3 = -3
-6Y + Y2 = -3

The Y term is -6Y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6Y + 9 + Y2 = -3 + 9

Reorder the terms:
9 + -6Y + Y2 = -3 + 9

Combine like terms: -3 + 9 = 6
9 + -6Y + Y2 = 6

Factor a perfect square on the left side:
(Y + -3)(Y + -3) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(Y + -3) equal to 2.449489743 and -2.449489743.

Subproblem 1

Y + -3 = 2.449489743 Simplifying Y + -3 = 2.449489743 Reorder the terms: -3 + Y = 2.449489743 Solving -3 + Y = 2.449489743 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + Y = 2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + Y = 2.449489743 + 3 Y = 2.449489743 + 3 Combine like terms: 2.449489743 + 3 = 5.449489743 Y = 5.449489743 Simplifying Y = 5.449489743

Subproblem 2

Y + -3 = -2.449489743 Simplifying Y + -3 = -2.449489743 Reorder the terms: -3 + Y = -2.449489743 Solving -3 + Y = -2.449489743 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + Y = -2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + Y = -2.449489743 + 3 Y = -2.449489743 + 3 Combine like terms: -2.449489743 + 3 = 0.550510257 Y = 0.550510257 Simplifying Y = 0.550510257

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {5.449489743, 0.550510257}

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