Question

Use the discriminant to determine the number of real solutions of x2+14x+24=0
.(

Answer 1

To determine the number of real solutions for the quadratic equation \( x^2 + 14x + 24 = 0 \) using the discriminant, we first identify the coefficients \( a \), \( b \), and \( c \) from the equation in the standard form \( ax^2 + bx + c = 0 \).

In this case:
- \( a = 1 \)
- \( b = 14 \)
- \( c = 24 \)

The discriminant \( D \) is calculated using the formula:

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

Substituting the values of \( a \), \( b \), and \( c \):

\[
D = (14)^2 - 4(1)(24)
\]
\[
D = 196 - 96
\]
\[
D = 100
\]

Now we can analyze the value of the discriminant:

- If \( D > 0 \), there are two distinct real solutions.
- If \( D = 0 \), there is exactly one real solution (a double root).
- If \( D < 0 \), there are no real solutions.

In this case, since \( D = 100 \) (which is greater than 0), we conclude that the quadratic equation \( x^2 + 14x + 24 = 0 \) has **two distinct real solutions**.