To determine the number of real solutions for the quadratic equation \( 25x^2 + 60x + 36 = 0 \), we can use the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
In this equation, \( a = 25 \), \( b = 60 \), and \( c = 36 \).
Let's calculate the discriminant:
\[ D = (60)^2 - 4(25)(36) \] \[ D = 3600 - 3600 \] \[ D = 0 \]
The discriminant \( D = 0 \) indicates that the quadratic equation has exactly one real solution (also known as a double root).
Thus, the number of real solutions for the equation \( 25x^2 + 60x + 36 = 0 \) is:
1 solution.