Question

How many real solutions does the following equation have?

25x^2 + 60x + 36 = 0

Answer 1

To determine how many real solutions the quadratic equation \( 25x^2 + 60x + 36 = 0 \) has, we can use the discriminant method. The discriminant (\( D \)) of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by the formula:

\[
D = b^2 - 4ac
\]

For our equation, the coefficients are:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)

Now, we calculate the discriminant:

\[
D = 60^2 - 4 \cdot 25 \cdot 36
\]

Calculating \( 60^2 \):

\[
60^2 = 3600
\]

Now calculating \( 4 \cdot 25 \cdot 36 \):

\[
4 \cdot 25 = 100
\]
\[
100 \cdot 36 = 3600
\]

Now substituting back into the formula for the discriminant:

\[
D = 3600 - 3600 = 0
\]

Since the discriminant \( D = 0 \), this indicates that the quadratic equation has exactly one real solution (a double root).

Therefore, the equation \( 25x^2 + 60x + 36 = 0 \) has:

\[
\boxed{1}
\]

real solution.