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 of the quadratic equation \(x^2 + 14x + 24 = 0\), we first need to calculate the discriminant, which is given by the formula:

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

In this equation, the coefficients are:
- \(a = 1\)
- \(b = 14\)
- \(c = 24\)

Now, substituting these values into the discriminant formula:

\[
D = 14^2 - 4 \cdot 1 \cdot 24
\]

Calculating \(14^2\):

\[
14^2 = 196
\]

Now calculating \(4 \cdot 1 \cdot 24\):

\[
4 \cdot 1 \cdot 24 = 96
\]

Now, substituting these values into the discriminant:

\[
D = 196 - 96 = 100
\]

Now we analyze the value of the discriminant \(D\):
- If \(D > 0\), there are two distinct real solutions.
- If \(D = 0\), there is exactly one real solution (a repeated root).
- If \(D < 0\), there are no real solutions.

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