y^2+3y=5

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Solution for y^2+3y=5 equation:


Simplifying
y2 + 3y = 5

Reorder the terms:
3y + y2 = 5

Solving
3y + y2 = 5

Solving for variable 'y'.

Reorder the terms:
-5 + 3y + y2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 3y + 5 + y2 = 0 + 5

Reorder the terms:
-5 + 5 + 3y + y2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 3y + y2 = 0 + 5
3y + y2 = 0 + 5

Combine like terms: 0 + 5 = 5
3y + y2 = 5

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 = 5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3y + y2 = 7.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + 1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

y + 1.5 = 2.692582404 Simplifying y + 1.5 = 2.692582404 Reorder the terms: 1.5 + y = 2.692582404 Solving 1.5 + y = 2.692582404 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 = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 2.692582404 + -1.5 y = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 y = 1.192582404 Simplifying y = 1.192582404

Subproblem 2

y + 1.5 = -2.692582404 Simplifying y + 1.5 = -2.692582404 Reorder the terms: 1.5 + y = -2.692582404 Solving 1.5 + y = -2.692582404 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 = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -2.692582404 + -1.5 y = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 y = -4.192582404 Simplifying y = -4.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.192582404, -4.192582404}

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