y^2+19y+24=0

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Solution for y^2+19y+24=0 equation: 


Simplifying
y2 + 19y + 24 = 0

Reorder the terms:
24 + 19y + y2 = 0

Solving
24 + 19y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + 19y + -24 + y2 = 0 + -24

Reorder the terms:
24 + -24 + 19y + y2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + 19y + y2 = 0 + -24
19y + y2 = 0 + -24

Combine like terms: 0 + -24 = -24
19y + y2 = -24

The y term is 19y.  Take half its coefficient (9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
19y + 90.25 + y2 = -24 + 90.25

Reorder the terms:
90.25 + 19y + y2 = -24 + 90.25

Combine like terms: -24 + 90.25 = 66.25
90.25 + 19y + y2 = 66.25

Factor a perfect square on the left side:
(y + 9.5)(y + 9.5) = 66.25

Calculate the square root of the right side: 8.139410298

Break this problem into two subproblems by setting 
(y + 9.5) equal to 8.139410298 and -8.139410298.


Subproblem 1
y + 9.5 = 8.139410298

Simplifying
y + 9.5 = 8.139410298

Reorder the terms:
9.5 + y = 8.139410298

Solving
9.5 + y = 8.139410298

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-9.5' to each side of the equation.
9.5 + -9.5 + y = 8.139410298 + -9.5

Combine like terms: 9.5 + -9.5 = 0.0
0.0 + y = 8.139410298 + -9.5
y = 8.139410298 + -9.5

Combine like terms: 8.139410298 + -9.5 = -1.360589702
y = -1.360589702

Simplifying
y = -1.360589702


Subproblem 2
y + 9.5 = -8.139410298

Simplifying
y + 9.5 = -8.139410298

Reorder the terms:
9.5 + y = -8.139410298

Solving
9.5 + y = -8.139410298

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-9.5' to each side of the equation.
9.5 + -9.5 + y = -8.139410298 + -9.5

Combine like terms: 9.5 + -9.5 = 0.0
0.0 + y = -8.139410298 + -9.5
y = -8.139410298 + -9.5

Combine like terms: -8.139410298 + -9.5 = -17.639410298
y = -17.639410298

Simplifying
y = -17.639410298


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-1.360589702, -17.639410298}

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