y^2+3y=38

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Solution for y^2+3y=38 equation:


Simplifying
y2 + 3y = 38

Reorder the terms:
3y + y2 = 38

Solving
3y + y2 = 38

Solving for variable 'y'.

Reorder the terms:
-38 + 3y + y2 = 38 + -38

Combine like terms: 38 + -38 = 0
-38 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + 3y + 38 + y2 = 0 + 38

Reorder the terms:
-38 + 38 + 3y + y2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + 3y + y2 = 0 + 38
3y + y2 = 0 + 38

Combine like terms: 0 + 38 = 38
3y + y2 = 38

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 = 38 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 38 + 2.25

Combine like terms: 38 + 2.25 = 40.25
2.25 + 3y + y2 = 40.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(y + 1.5) equal to 6.34428877 and -6.34428877.

Subproblem 1

y + 1.5 = 6.34428877 Simplifying y + 1.5 = 6.34428877 Reorder the terms: 1.5 + y = 6.34428877 Solving 1.5 + y = 6.34428877 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 = 6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 6.34428877 + -1.5 y = 6.34428877 + -1.5 Combine like terms: 6.34428877 + -1.5 = 4.84428877 y = 4.84428877 Simplifying y = 4.84428877

Subproblem 2

y + 1.5 = -6.34428877 Simplifying y + 1.5 = -6.34428877 Reorder the terms: 1.5 + y = -6.34428877 Solving 1.5 + y = -6.34428877 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 = -6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -6.34428877 + -1.5 y = -6.34428877 + -1.5 Combine like terms: -6.34428877 + -1.5 = -7.84428877 y = -7.84428877 Simplifying y = -7.84428877

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.84428877, -7.84428877}

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