y^2=-3y+9

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Solution for y^2=-3y+9 equation:


Simplifying
y2 = -3y + 9

Reorder the terms:
y2 = 9 + -3y

Solving
y2 = 9 + -3y

Solving for variable 'y'.

Reorder the terms:
-9 + 3y + y2 = 9 + -3y + -9 + 3y

Reorder the terms:
-9 + 3y + y2 = 9 + -9 + -3y + 3y

Combine like terms: 9 + -9 = 0
-9 + 3y + y2 = 0 + -3y + 3y
-9 + 3y + y2 = -3y + 3y

Combine like terms: -3y + 3y = 0
-9 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 3y + 9 + y2 = 0 + 9

Reorder the terms:
-9 + 9 + 3y + y2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 3y + y2 = 0 + 9
3y + y2 = 0 + 9

Combine like terms: 0 + 9 = 9
3y + y2 = 9

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 9 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 9 + 2.25

Combine like terms: 9 + 2.25 = 11.25
2.25 + 3y + y2 = 11.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(y + 1.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

y + 1.5 = 3.354101966 Simplifying y + 1.5 = 3.354101966 Reorder the terms: 1.5 + y = 3.354101966 Solving 1.5 + y = 3.354101966 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 3.354101966 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 3.354101966 + -1.5 y = 3.354101966 + -1.5 Combine like terms: 3.354101966 + -1.5 = 1.854101966 y = 1.854101966 Simplifying y = 1.854101966

Subproblem 2

y + 1.5 = -3.354101966 Simplifying y + 1.5 = -3.354101966 Reorder the terms: 1.5 + y = -3.354101966 Solving 1.5 + y = -3.354101966 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -3.354101966 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -3.354101966 + -1.5 y = -3.354101966 + -1.5 Combine like terms: -3.354101966 + -1.5 = -4.854101966 y = -4.854101966 Simplifying y = -4.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.854101966, -4.854101966}

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