Question

For the function ​f(x)equals4 x squared minus 17 x minus 15​,
a. Calculate the discriminant.
b. Determine whether there are​ 0, 1, or 2 real solutions to ​f(x)equals0.
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Part 1
a. The discriminant is
  
enter your response here. ​(Simplify your​ answer.)

Answer 1

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 \( f(x) = 0 \).