Chapter 3.3, Problem 11AYU

If $b^2-4ac>0$, which of the following conclusions can be made about the graph of $f(x)=a{x}^2+bx+c,a\ne 0$?

(a) The graph has two distinct $x\text{-intercepts}$.

(b) The graph has no $x\text{-intercepts}$.

(c) The graph has three distinct $x\text{-intercepts}$.

(d) The graph has one $x\text{-intercepts}$.

To determine

Which of the given statements is a conclusion.

Answer to Problem 11AYU

Thus statement (a) follows the given conclusion.

(a) The graph of the given function will have 2 distinct $x\text{-intercepts}$.

Explanation of Solution

Given:

It is given that $b^2-4ac>0$

Consider the quadratic equation $f(x)=a{x}^2+bx+c,a\ne 0$

We can find the solution to the given equation by using the quadratic formula.

Therefore, we can find the solution as

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

If the discriminant $b^2-4ac=k>0,$ then we get the solution as

$x=\frac{-b+k}{2a},<mtext> </mtext>\frac{-b-k}{2a}$

Therefore, we can see that the graph of the given function will have 2 distinct $x\text{-intercepts}$.