To find the discriminant of the quadratic function \( f(x) = 4x^2 - 17x - 15 \), we can use the formula for the discriminant \( D \) given by:
\[ D = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
For the given function:
- \( a = 4 \)
- \( b = -17 \)
- \( c = -15 \)
Now, we can calculate the discriminant:
\[ D = (-17)^2 - 4 \cdot 4 \cdot (-15) \]
Calculating \( (-17)^2 \):
\[ D = 289 - 4 \cdot 4 \cdot (-15) \]
Calculating \( 4 \cdot 4 \cdot (-15) \):
\[ 4 \cdot 4 = 16 \quad \text{then} \quad 16 \cdot (-15) = -240 \]
So,
\[ D = 289 - (-240) = 289 + 240 = 529 \]
Now, we have:
\[ \text{The discriminant is } 529 \]
Part 2: Determine the number of real solutions
The number of real solutions is determined by the value of the discriminant \( D \):
- If \( D > 0 \), there are 2 real solutions.
- If \( D = 0 \), there is 1 real solution.
- If \( D < 0 \), there are 0 real solutions.
Since \( D = 529 \) which is greater than 0, we conclude that there are 2 real solutions to \( f(x) = 0 \).
Summary:
- a. The discriminant is \( 529 \).
- b. There are 2 real solutions to \( f(x) = 0 \).