y^2-6y+9=6

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Solution for y^2-6y+9=6 equation:


Simplifying
y2 + -6y + 9 = 6

Reorder the terms:
9 + -6y + y2 = 6

Solving
9 + -6y + y2 = 6

Solving for variable 'y'.

Reorder the terms:
9 + -6 + -6y + y2 = 6 + -6

Combine like terms: 9 + -6 = 3
3 + -6y + y2 = 6 + -6

Combine like terms: 6 + -6 = 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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