Consider the following function.

[tex]\[ f(x) = x^2 - 7x \][/tex]

(a) Use a graphing utility to graph the function and find the zeros of the function. (Enter your answers as a comma-separated list.)

[tex]\[ x = \square \][/tex]

(b) Verify your results from part (a) algebraically. (Enter your answers as a comma-separated list.)

[tex]\[ x = \square \][/tex]



Answer :

Let's solve the problem step-by-step.

### Part (a) - Finding the zeros of the function

To find the zeros (roots) of the function [tex]\( f(x) = x^2 - 7x \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].

Set [tex]\( f(x) \)[/tex] equal to zero:
[tex]\[ x^2 - 7x = 0 \][/tex]

Factor the quadratic equation:
[tex]\[ x(x - 7) = 0 \][/tex]

Set each factor equal to zero:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x - 7 = 0 \)[/tex]

Solving these for [tex]\( x \)[/tex] gives us:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = 7 \][/tex]

So, the zeros of the function are:
[tex]\[ x = 0, 7 \][/tex]

### Part (b) - Verifying the results algebraically

To verify the results, we will use the same equation and factor it.

Start with the original function:
[tex]\[ f(x) = x^2 - 7x \][/tex]

We already factored it in part (a):
[tex]\[ x(x - 7) = 0 \][/tex]

Setting each factor to zero, we get:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x = 7 \)[/tex]

Thus, the zeros of the function are confirmed once again to be:
[tex]\[ x = 0, 7 \][/tex]

Hence, the zeros of the function [tex]\( f(x) = x^2 - 7x \)[/tex] are indeed [tex]\( x = 0 \)[/tex] and [tex]\( x = 7 \)[/tex].

The factorization also agrees with this:
[tex]\[ x^2 - 7x = x(x - 7) \][/tex]

Now let's summarize the answers:

(a) The zeros of the function are:
[tex]\[ x = 0, 7 \][/tex]

(b) Verified results algebraically:
[tex]\[ x = 0, 7 \][/tex]