5. \( x^{2}+6 x+9=0 \)

This equation is a perfect square trinomial, which can be factored as \( (x + 3)^2 = 0 \). This means that the solution to the equation is \( x = -3 \) with a multiplicity of 2, indicating that the graph of the related function touches the x-axis at this point but does not cross it. When solving quadratic equations like this, a common mistake is forgetting to check the discriminant. For \( ax^2 + bx + c = 0 \), you calculate the discriminant with \( D = b^2 - 4ac \). If \( D < 0 \), the equation has no real solutions, and if \( D = 0 \), it has one real solution (like in this case!). Always double-check to avoid overlooking these scenarios!