y^2+4y=41

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Solution for y^2+4y=41 equation:


Simplifying
y2 + 4y = 41

Reorder the terms:
4y + y2 = 41

Solving
4y + y2 = 41

Solving for variable 'y'.

Reorder the terms:
-41 + 4y + y2 = 41 + -41

Combine like terms: 41 + -41 = 0
-41 + 4y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + 4y + 41 + y2 = 0 + 41

Reorder the terms:
-41 + 41 + 4y + y2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + 4y + y2 = 0 + 41
4y + y2 = 0 + 41

Combine like terms: 0 + 41 = 41
4y + y2 = 41

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = 41 + 4

Reorder the terms:
4 + 4y + y2 = 41 + 4

Combine like terms: 41 + 4 = 45
4 + 4y + y2 = 45

Factor a perfect square on the left side:
(y + 2)(y + 2) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(y + 2) equal to 6.708203933 and -6.708203933.

Subproblem 1

y + 2 = 6.708203933 Simplifying y + 2 = 6.708203933 Reorder the terms: 2 + y = 6.708203933 Solving 2 + y = 6.708203933 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = 6.708203933 + -2 Combine like terms: 2 + -2 = 0 0 + y = 6.708203933 + -2 y = 6.708203933 + -2 Combine like terms: 6.708203933 + -2 = 4.708203933 y = 4.708203933 Simplifying y = 4.708203933

Subproblem 2

y + 2 = -6.708203933 Simplifying y + 2 = -6.708203933 Reorder the terms: 2 + y = -6.708203933 Solving 2 + y = -6.708203933 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = -6.708203933 + -2 Combine like terms: 2 + -2 = 0 0 + y = -6.708203933 + -2 y = -6.708203933 + -2 Combine like terms: -6.708203933 + -2 = -8.708203933 y = -8.708203933 Simplifying y = -8.708203933

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.708203933, -8.708203933}

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