To determine the number of real solutions of the quadratic equation \( 25x^2 + 60x + 36 = 0 \), we need to calculate the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
For the equation \( 25x^2 + 60x + 36 = 0 \):
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Now we compute the discriminant \( D \):
\[ D = b^2 - 4ac = 60^2 - 4 \cdot 25 \cdot 36 \]
Calculating \( b^2 \):
\[ 60^2 = 3600 \]
Calculating \( 4ac \):
\[ 4 \cdot 25 \cdot 36 = 100 \cdot 36 = 3600 \]
Now substituting these values back into the discriminant formula:
\[ D = 3600 - 3600 = 0 \]
Since the discriminant \( D = 0 \), this indicates that the quadratic equation has exactly one real solution (a double root).
Thus, the number of real solutions to the equation \( 25x^2 + 60x + 36 = 0 \) is:
\[ \boxed{1} \]