y^2=6y+3

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Solution for y^2=6y+3 equation:


Simplifying
y2 = 6y + 3

Reorder the terms:
y2 = 3 + 6y

Solving
y2 = 3 + 6y

Solving for variable 'y'.

Reorder the terms:
-3 + -6y + y2 = 3 + 6y + -3 + -6y

Reorder the terms:
-3 + -6y + y2 = 3 + -3 + 6y + -6y

Combine like terms: 3 + -3 = 0
-3 + -6y + y2 = 0 + 6y + -6y
-3 + -6y + y2 = 6y + -6y

Combine like terms: 6y + -6y = 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 = 12
9 + -6y + y2 = 12

Factor a perfect square on the left side:
(y + -3)(y + -3) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(y + -3) equal to 3.464101615 and -3.464101615.

Subproblem 1

y + -3 = 3.464101615 Simplifying y + -3 = 3.464101615 Reorder the terms: -3 + y = 3.464101615 Solving -3 + y = 3.464101615 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 = 3.464101615 + 3 Combine like terms: -3 + 3 = 0 0 + y = 3.464101615 + 3 y = 3.464101615 + 3 Combine like terms: 3.464101615 + 3 = 6.464101615 y = 6.464101615 Simplifying y = 6.464101615

Subproblem 2

y + -3 = -3.464101615 Simplifying y + -3 = -3.464101615 Reorder the terms: -3 + y = -3.464101615 Solving -3 + y = -3.464101615 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 = -3.464101615 + 3 Combine like terms: -3 + 3 = 0 0 + y = -3.464101615 + 3 y = -3.464101615 + 3 Combine like terms: -3.464101615 + 3 = -0.464101615 y = -0.464101615 Simplifying y = -0.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. y = {6.464101615, -0.464101615}

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