3q(3q)+16q=-5

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Solution for 3q(3q)+16q=-5 equation:


Simplifying
3q(3q) + 16q = -5

Remove parenthesis around (3q)
3q * 3q + 16q = -5

Reorder the terms for easier multiplication:
3 * 3q * q + 16q = -5

Multiply 3 * 3
9q * q + 16q = -5

Multiply q * q
9q2 + 16q = -5

Reorder the terms:
16q + 9q2 = -5

Solving
16q + 9q2 = -5

Solving for variable 'q'.

Reorder the terms:
5 + 16q + 9q2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + 16q + 9q2 = 0

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

Divide each side by '9'.
0.5555555556 + 1.777777778q + q2 = 0

Move the constant term to the right:

Add '-0.5555555556' to each side of the equation.
0.5555555556 + 1.777777778q + -0.5555555556 + q2 = 0 + -0.5555555556

Reorder the terms:
0.5555555556 + -0.5555555556 + 1.777777778q + q2 = 0 + -0.5555555556

Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000
0.0000000000 + 1.777777778q + q2 = 0 + -0.5555555556
1.777777778q + q2 = 0 + -0.5555555556

Combine like terms: 0 + -0.5555555556 = -0.5555555556
1.777777778q + q2 = -0.5555555556

The q term is 1.777777778q.  Take half its coefficient (0.888888889).
Square it (0.7901234570) and add it to both sides.

Add '0.7901234570' to each side of the equation.
1.777777778q + 0.7901234570 + q2 = -0.5555555556 + 0.7901234570

Reorder the terms:
0.7901234570 + 1.777777778q + q2 = -0.5555555556 + 0.7901234570

Combine like terms: -0.5555555556 + 0.7901234570 = 0.2345679014
0.7901234570 + 1.777777778q + q2 = 0.2345679014

Factor a perfect square on the left side:
(q + 0.888888889)(q + 0.888888889) = 0.2345679014

Calculate the square root of the right side: 0.484322105

Break this problem into two subproblems by setting 
(q + 0.888888889) equal to 0.484322105 and -0.484322105.

Subproblem 1

q + 0.888888889 = 0.484322105 Simplifying q + 0.888888889 = 0.484322105 Reorder the terms: 0.888888889 + q = 0.484322105 Solving 0.888888889 + q = 0.484322105 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + q = 0.484322105 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + q = 0.484322105 + -0.888888889 q = 0.484322105 + -0.888888889 Combine like terms: 0.484322105 + -0.888888889 = -0.404566784 q = -0.404566784 Simplifying q = -0.404566784

Subproblem 2

q + 0.888888889 = -0.484322105 Simplifying q + 0.888888889 = -0.484322105 Reorder the terms: 0.888888889 + q = -0.484322105 Solving 0.888888889 + q = -0.484322105 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + q = -0.484322105 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + q = -0.484322105 + -0.888888889 q = -0.484322105 + -0.888888889 Combine like terms: -0.484322105 + -0.888888889 = -1.373210994 q = -1.373210994 Simplifying q = -1.373210994

Solution

The solution to the problem is based on the solutions from the subproblems. q = {-0.404566784, -1.373210994}

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