3y=y^2-7

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Solution for 3y=y^2-7 equation:


Simplifying
3y = y2 + -7

Reorder the terms:
3y = -7 + y2

Solving
3y = -7 + y2

Solving for variable 'y'.

Reorder the terms:
7 + 3y + -1y2 = -7 + y2 + 7 + -1y2

Reorder the terms:
7 + 3y + -1y2 = -7 + 7 + y2 + -1y2

Combine like terms: -7 + 7 = 0
7 + 3y + -1y2 = 0 + y2 + -1y2
7 + 3y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
7 + 3y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + -3y + y2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -3y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + -3y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -3y + y2 = 0 + 7
-3y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
-3y + y2 = 7

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 = 7 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + -3y + y2 = 9.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(y + -1.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

y + -1.5 = 3.041381265 Simplifying y + -1.5 = 3.041381265 Reorder the terms: -1.5 + y = 3.041381265 Solving -1.5 + y = 3.041381265 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 = 3.041381265 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 3.041381265 + 1.5 y = 3.041381265 + 1.5 Combine like terms: 3.041381265 + 1.5 = 4.541381265 y = 4.541381265 Simplifying y = 4.541381265

Subproblem 2

y + -1.5 = -3.041381265 Simplifying y + -1.5 = -3.041381265 Reorder the terms: -1.5 + y = -3.041381265 Solving -1.5 + y = -3.041381265 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 = -3.041381265 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -3.041381265 + 1.5 y = -3.041381265 + 1.5 Combine like terms: -3.041381265 + 1.5 = -1.541381265 y = -1.541381265 Simplifying y = -1.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.541381265, -1.541381265}

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