q^2+6q-19=0

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Solution for q^2+6q-19=0 equation:


Simplifying
q2 + 6q + -19 = 0

Reorder the terms:
-19 + 6q + q2 = 0

Solving
-19 + 6q + q2 = 0

Solving for variable 'q'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 6q + 19 + q2 = 0 + 19

Reorder the terms:
-19 + 19 + 6q + q2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 6q + q2 = 0 + 19
6q + q2 = 0 + 19

Combine like terms: 0 + 19 = 19
6q + q2 = 19

The q term is 6q.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6q + 9 + q2 = 19 + 9

Reorder the terms:
9 + 6q + q2 = 19 + 9

Combine like terms: 19 + 9 = 28
9 + 6q + q2 = 28

Factor a perfect square on the left side:
(q + 3)(q + 3) = 28

Calculate the square root of the right side: 5.291502622

Break this problem into two subproblems by setting 
(q + 3) equal to 5.291502622 and -5.291502622.

Subproblem 1

q + 3 = 5.291502622 Simplifying q + 3 = 5.291502622 Reorder the terms: 3 + q = 5.291502622 Solving 3 + q = 5.291502622 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = 5.291502622 + -3 Combine like terms: 3 + -3 = 0 0 + q = 5.291502622 + -3 q = 5.291502622 + -3 Combine like terms: 5.291502622 + -3 = 2.291502622 q = 2.291502622 Simplifying q = 2.291502622

Subproblem 2

q + 3 = -5.291502622 Simplifying q + 3 = -5.291502622 Reorder the terms: 3 + q = -5.291502622 Solving 3 + q = -5.291502622 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = -5.291502622 + -3 Combine like terms: 3 + -3 = 0 0 + q = -5.291502622 + -3 q = -5.291502622 + -3 Combine like terms: -5.291502622 + -3 = -8.291502622 q = -8.291502622 Simplifying q = -8.291502622

Solution

The solution to the problem is based on the solutions from the subproblems. q = {2.291502622, -8.291502622}

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