Question

How many real solutions does the following quadratic equation have?

25x2+60x+36=0

Answer 1

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**.