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