To find the zeros of the function [tex]\( f(x) = x^2 - 11x + 24 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. Here's a step-by-step solution:
1. Write down the equation:
[tex]\[
x^2 - 11x + 24 = 0
\][/tex]
2. Factor the quadratic expression:
To factor [tex]\( x^2 - 11x + 24 \)[/tex], we need to find two numbers that multiply to the constant term (24) and add up to the coefficient of the linear term (-11).
3. Find the factors of 24 that add up to -11:
The pairs of factors of 24 are:
- [tex]\(1 \cdot 24\)[/tex]
- [tex]\(2 \cdot 12\)[/tex]
- [tex]\(3 \cdot 8\)[/tex]
- [tex]\(4 \cdot 6\)[/tex]
Among these, the pair [tex]\(3\)[/tex] and [tex]\(8\)[/tex] satisfy the condition because:
[tex]\[
3 + 8 = 11 \quad \text{and} \quad 3 \cdot 8 = 24
\][/tex]
4. Express the middle term using these factors:
Rewriting [tex]\( -11x \)[/tex] as [tex]\( -3x - 8x \)[/tex]:
[tex]\[
x^2 - 3x - 8x + 24 = 0
\][/tex]
5. Factor by grouping:
Group the terms:
[tex]\[
(x^2 - 3x) - (8x - 24) = 0
\][/tex]
Factor out the common factors in each group:
[tex]\[
x(x - 3) - 8(x - 3) = 0
\][/tex]
Notice that [tex]\( (x - 3) \)[/tex] is a common factor:
[tex]\[
(x - 3)(x - 8) = 0
\][/tex]
6. Solve for the zeros:
Set each factor equal to zero:
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
[tex]\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\][/tex]
Thus, the zeros of the function [tex]\( f(x) = x^2 - 11x + 24 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = 8 \)[/tex].
7. Choose the correct answer:
The correct answer is:
[tex]\[
\text{D. } \quad x = 3 ; 8
\][/tex]
