To determine the number of real solutions of the quadratic equation \(x^2 + 14x + 24 = 0\) using the discriminant, we first identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\):
- \(a = 1\)
- \(b = 14\)
- \(c = 24\)
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Now, we can substitute the values of \(a\), \(b\), and \(c\):
\[ D = (14)^2 - 4(1)(24) \]
Calculating further:
\[ D = 196 - 96 \] \[ D = 100 \]
Now, we analyze the value of the discriminant:
- If \(D > 0\), there are 2 distinct real solutions.
- If \(D = 0\), there is 1 real solution (a repeated root).
- If \(D < 0\), there are no real solutions.
Since \(D = 100\) which is greater than 0, we conclude that there are 2 distinct real solutions for the equation \(x^2 + 14x + 24 = 0\).