To determine the number of real solutions of the quadratic equation \( 25x^2 + 60x + 36 = 0 \), we can use the discriminant method.
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Now, we can substitute \( a \), \( b \), and \( c \) into this formula:
\[ D = 60^2 - 4 \cdot 25 \cdot 36 \]
Calculating \( 60^2 \):
\[ 60^2 = 3600 \]
Now calculating \( 4 \cdot 25 \cdot 36 \):
\[ 4 \cdot 25 \cdot 36 = 100 \cdot 36 = 3600 \]
Putting it all together:
\[ D = 3600 - 3600 = 0 \]
The discriminant \( D = 0 \) indicates that the quadratic equation has exactly one real solution (also known as a double root).
Thus, the quadratic equation \( 25x^2 + 60x + 36 = 0 \) has one real solution.