or the function ​f(x)equals3 x squared minus 17 x minus 6​,

Question

a. Calculate the discriminant.
b. Determine whether there are​ 0, 1, or 2 real solutions to ​f(x)equals0.
Question content area bottom
Part 1
a. The discriminant is
  
enter your response here. ​(Simplify your​ answer.)

Answer 1

To find the discriminant of the quadratic function \( f(x) = 3x^2 - 17x - 6 \), we will use the formula for the discriminant, which is given by:

\[ D = b^2 - 4ac \]

where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation of the form \( ax^2 + bx + c \).

In your case:

• \( a = 3 \)
• \( b = -17 \)
• \( c = -6 \)

Now, we can plug these values into the discriminant formula:

\[ D = (-17)^2 - 4 \cdot 3 \cdot (-6) \] \[ D = 289 + 72 \] \[ D = 289 + 72 = 361 \]

So, the discriminant is:

\[ \boxed{361} \]

---

Part 2: Determine the number of real solutions

Now, to determine whether there are 0, 1, or 2 real solutions to \( f(x) = 0 \):

• 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 the discriminant \( D = 361 \) is greater than 0, we conclude that there are:

\[ \text{2 real solutions.} \]