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.