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

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.