q^2+3q+1=0

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Solution for q^2+3q+1=0 equation:


Simplifying
q2 + 3q + 1 = 0

Reorder the terms:
1 + 3q + q2 = 0

Solving
1 + 3q + q2 = 0

Solving for variable 'q'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3q + -1 + q2 = 0 + -1

Reorder the terms:
1 + -1 + 3q + q2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3q + q2 = 0 + -1
3q + q2 = 0 + -1

Combine like terms: 0 + -1 = -1
3q + q2 = -1

The q term is 3q.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3q + 2.25 + q2 = -1 + 2.25

Reorder the terms:
2.25 + 3q + q2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3q + q2 = 1.25

Factor a perfect square on the left side:
(q + 1.5)(q + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(q + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

q + 1.5 = 1.118033989 Simplifying q + 1.5 = 1.118033989 Reorder the terms: 1.5 + q = 1.118033989 Solving 1.5 + q = 1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = 1.118033989 + -1.5 q = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 q = -0.381966011 Simplifying q = -0.381966011

Subproblem 2

q + 1.5 = -1.118033989 Simplifying q + 1.5 = -1.118033989 Reorder the terms: 1.5 + q = -1.118033989 Solving 1.5 + q = -1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = -1.118033989 + -1.5 q = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 q = -2.618033989 Simplifying q = -2.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. q = {-0.381966011, -2.618033989}

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