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Find the zeros of the following function:

[tex]\[ f(x) = x^2 - 11x + 24 \][/tex]

A. [tex]\( x = 6, 5 \)[/tex]
B. [tex]\( x = 1, 24 \)[/tex]
C. [tex]\( x = -8, -3 \)[/tex]
D. [tex]\( x = 3, 8 \)[/tex]


Sagot :

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]