Final answer:
The quadratic equation u²+18u+81=0 is a perfect square trinomial and can be factored as (u+9)² = 0, which gives the solution u = -9.
Step-by-step explanation:
To solve the equation u²+18u+81=0 using perfect square factoring patterns, we can recognize that the quadratic is already in the form of a perfect square trinomial. A perfect square trinomial is of the form (a+b)² = a² + 2ab + b². Here, 81 is a perfect square (9²), and 18u is twice the product of 9 and u, implying that the expression is a perfect square. Following this pattern, we can rewrite the equation as (u+9)² = 0.
Setting the square equal to zero gives us u + 9 = 0. Solving for u, we subtract 9 from both sides to get u = -9. The solution to the equation is u = -9.