s^2+24s+31=0

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Solution for s^2+24s+31=0 equation:


Simplifying
s2 + 24s + 31 = 0

Reorder the terms:
31 + 24s + s2 = 0

Solving
31 + 24s + s2 = 0

Solving for variable 's'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 24s + -31 + s2 = 0 + -31

Reorder the terms:
31 + -31 + 24s + s2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 24s + s2 = 0 + -31
24s + s2 = 0 + -31

Combine like terms: 0 + -31 = -31
24s + s2 = -31

The s term is 24s.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24s + 144 + s2 = -31 + 144

Reorder the terms:
144 + 24s + s2 = -31 + 144

Combine like terms: -31 + 144 = 113
144 + 24s + s2 = 113

Factor a perfect square on the left side:
(s + 12)(s + 12) = 113

Calculate the square root of the right side: 10.630145813

Break this problem into two subproblems by setting 
(s + 12) equal to 10.630145813 and -10.630145813.

Subproblem 1

s + 12 = 10.630145813 Simplifying s + 12 = 10.630145813 Reorder the terms: 12 + s = 10.630145813 Solving 12 + s = 10.630145813 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + s = 10.630145813 + -12 Combine like terms: 12 + -12 = 0 0 + s = 10.630145813 + -12 s = 10.630145813 + -12 Combine like terms: 10.630145813 + -12 = -1.369854187 s = -1.369854187 Simplifying s = -1.369854187

Subproblem 2

s + 12 = -10.630145813 Simplifying s + 12 = -10.630145813 Reorder the terms: 12 + s = -10.630145813 Solving 12 + s = -10.630145813 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + s = -10.630145813 + -12 Combine like terms: 12 + -12 = 0 0 + s = -10.630145813 + -12 s = -10.630145813 + -12 Combine like terms: -10.630145813 + -12 = -22.630145813 s = -22.630145813 Simplifying s = -22.630145813

Solution

The solution to the problem is based on the solutions from the subproblems. s = {-1.369854187, -22.630145813}

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