Question

find the x-intercepts, if they exist.
$f(x)=5x^2 - x + 20$
find the x-intercepts. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercept(s) is(are)
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there is no x-intercept.

Explanation:

Step 1: Recall the formula for the discriminant of a quadratic function

For a quadratic function of the form \(f(x)=ax^{2}+bx + c\) (where \(a
eq0\)), the discriminant \(\Delta\) is given by the formula \(\Delta=b^{2}-4ac\). In the function \(f(x) = 5x^{2}-x + 20\), we have \(a = 5\), \(b=- 1\) and \(c = 20\).

Step 2: Calculate the discriminant

Substitute the values of \(a\), \(b\) and \(c\) into the discriminant formula:
\(\Delta=(-1)^{2}-4\times5\times20\)
First, calculate \((-1)^{2}=1\) and \(4\times5\times20 = 400\).
Then, \(\Delta=1 - 400=- 399\)

Step 3: Analyze the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\), the number of real roots (which correspond to the x - intercepts of the function \(y = ax^{2}+bx + c\)) is determined by the discriminant:

• If \(\Delta>0\), there are two distinct real roots (two x - intercepts).
• If \(\Delta = 0\), there is one real root (one x - intercept).
• If \(\Delta<0\), there are no real roots (no x - intercepts).

Since \(\Delta=-399<0\), the quadratic equation \(5x^{2}-x + 20 = 0\) has no real solutions. So the function \(f(x)=5x^{2}-x + 20\) has no x - intercepts.

Answer:

B. There is no x - intercept.