3s^2+16s+15=0

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Solution for 3s^2+16s+15=0 equation:


Simplifying
3s2 + 16s + 15 = 0

Reorder the terms:
15 + 16s + 3s2 = 0

Solving
15 + 16s + 3s2 = 0

Solving for variable 's'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
5 + 5.333333333s + s2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 5.333333333s + -5 + s2 = 0 + -5

Reorder the terms:
5 + -5 + 5.333333333s + s2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 5.333333333s + s2 = 0 + -5
5.333333333s + s2 = 0 + -5

Combine like terms: 0 + -5 = -5
5.333333333s + s2 = -5

The s term is 5.333333333s.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333s + 7.111111113 + s2 = -5 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333s + s2 = -5 + 7.111111113

Combine like terms: -5 + 7.111111113 = 2.111111113
7.111111113 + 5.333333333s + s2 = 2.111111113

Factor a perfect square on the left side:
(s + 2.666666667)(s + 2.666666667) = 2.111111113

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(s + 2.666666667) equal to 1.452966315 and -1.452966315.

Subproblem 1

s + 2.666666667 = 1.452966315 Simplifying s + 2.666666667 = 1.452966315 Reorder the terms: 2.666666667 + s = 1.452966315 Solving 2.666666667 + s = 1.452966315 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + s = 1.452966315 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + s = 1.452966315 + -2.666666667 s = 1.452966315 + -2.666666667 Combine like terms: 1.452966315 + -2.666666667 = -1.213700352 s = -1.213700352 Simplifying s = -1.213700352

Subproblem 2

s + 2.666666667 = -1.452966315 Simplifying s + 2.666666667 = -1.452966315 Reorder the terms: 2.666666667 + s = -1.452966315 Solving 2.666666667 + s = -1.452966315 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + s = -1.452966315 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + s = -1.452966315 + -2.666666667 s = -1.452966315 + -2.666666667 Combine like terms: -1.452966315 + -2.666666667 = -4.119632982 s = -4.119632982 Simplifying s = -4.119632982

Solution

The solution to the problem is based on the solutions from the subproblems. s = {-1.213700352, -4.119632982}

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