To determine the number of real solutions of the quadratic equation \(25x^2 + 60x + 36 = 0\), we can use the discriminant method. The discriminant \(\Delta\) for a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by the formula:
\[ \Delta = b^2 - 4ac \]
In this case, the coefficients are:
- \(a = 25\)
- \(b = 60\)
- \(c = 36\)
Now, we can calculate the discriminant:
\[ \Delta = 60^2 - 4 \cdot 25 \cdot 36 \]
Calculating \(60^2\):
\[ 60^2 = 3600 \]
Next, calculating \(4 \cdot 25 \cdot 36\):
\[ 4 \cdot 25 = 100 \]
\[ 100 \cdot 36 = 3600 \]
Now, substituting back into the discriminant:
\[ \Delta = 3600 - 3600 = 0 \]
Since the discriminant \(\Delta\) is equal to 0, the quadratic equation has exactly one real solution (a double root).
Therefore, the answer is:
1 real solution.