To determine the number of real solutions of the quadratic equation \( 25x^2 + 60x + 36 = 0 \), we can use the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
Here, we have:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Now, we can calculate the discriminant:
\[ D = 60^2 - 4 \cdot 25 \cdot 36 \] \[ D = 3600 - 3600 \] \[ D = 0 \]
The discriminant \( D = 0 \) indicates that the quadratic equation has exactly one real solution (also known as a double root).
Therefore, the number of real solutions to the equation \( 25x^2 + 60x + 36 = 0 \) is 1.
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