To determine how many real solutions the quadratic equation \( 25x^2 + 60x + 36 = 0 \) has, we can use the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
For the given equation, we have:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Now, we can calculate the discriminant:
\[ D = 60^2 - 4 \cdot 25 \cdot 36 \]
Calculating \( 60^2 \):
\[ 60^2 = 3600 \]
Calculating \( 4 \cdot 25 \cdot 36 \):
\[ 4 \cdot 25 = 100 \] \[ 100 \cdot 36 = 3600 \]
Now substituting back into the discriminant:
\[ D = 3600 - 3600 = 0 \]
The discriminant \( D = 0 \) indicates that there is exactly one real solution (also known as a repeated or double root).
Therefore, the quadratic equation \( 25x^2 + 60x + 36 = 0 \) has 1 real solution.
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