y^2+20y=-52

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Solution for y^2+20y=-52 equation:


Simplifying
y2 + 20y = -52

Reorder the terms:
20y + y2 = -52

Solving
20y + y2 = -52

Solving for variable 'y'.

Reorder the terms:
52 + 20y + y2 = -52 + 52

Combine like terms: -52 + 52 = 0
52 + 20y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-52' to each side of the equation.
52 + 20y + -52 + y2 = 0 + -52

Reorder the terms:
52 + -52 + 20y + y2 = 0 + -52

Combine like terms: 52 + -52 = 0
0 + 20y + y2 = 0 + -52
20y + y2 = 0 + -52

Combine like terms: 0 + -52 = -52
20y + y2 = -52

The y term is 20y.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20y + 100 + y2 = -52 + 100

Reorder the terms:
100 + 20y + y2 = -52 + 100

Combine like terms: -52 + 100 = 48
100 + 20y + y2 = 48

Factor a perfect square on the left side:
(y + 10)(y + 10) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(y + 10) equal to 6.92820323 and -6.92820323.

Subproblem 1

y + 10 = 6.92820323 Simplifying y + 10 = 6.92820323 Reorder the terms: 10 + y = 6.92820323 Solving 10 + y = 6.92820323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + y = 6.92820323 + -10 Combine like terms: 10 + -10 = 0 0 + y = 6.92820323 + -10 y = 6.92820323 + -10 Combine like terms: 6.92820323 + -10 = -3.07179677 y = -3.07179677 Simplifying y = -3.07179677

Subproblem 2

y + 10 = -6.92820323 Simplifying y + 10 = -6.92820323 Reorder the terms: 10 + y = -6.92820323 Solving 10 + y = -6.92820323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + y = -6.92820323 + -10 Combine like terms: 10 + -10 = 0 0 + y = -6.92820323 + -10 y = -6.92820323 + -10 Combine like terms: -6.92820323 + -10 = -16.92820323 y = -16.92820323 Simplifying y = -16.92820323

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-3.07179677, -16.92820323}

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