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]
