Question

or the function ​f(x)equals3 x squared minus 17 x minus 6​,
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 \( f(x) = 3x^2 - 17x - 6 \), 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:

\[
\boxed{361}
\]

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### 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:

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