To find the solutions of the equation z² + 12z + 36 = 0, we can factor the quadratic expression. The equation can be rewritten as:
(z + 6)(z + 6) = 0
or simply:
(z + 6)² = 0
From this factorization, we can see that the only solution occurs when:
z + 6 = 0
Solving for z gives:
z = -6
Thus, the equation has a repeated root or a double solution. The only solution is:
z = -6