To determine the number of real solutions for the quadratic equation \( 25x^2 + 60x + 36 = 0 \), we can use the discriminant. The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
In this equation, we have:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Now, we can calculate the discriminant:
\[ D = (60)^2 - 4(25)(36) \]
Calculating \( (60)^2 \):
\[ (60)^2 = 3600 \]
Next, calculating \( 4(25)(36) \):
\[ 4(25)(36) = 4 \times 25 \times 36 = 100 \times 36 = 3600 \]
Now we substitute these values into the equation for the discriminant:
\[ D = 3600 - 3600 = 0 \]
Since the discriminant \( D = 0 \), this indicates that the quadratic equation has exactly one real solution (a repeated root).
Thus, the final answer is:
1 solution.