What are the zeros of [tex]$f(x)=x^2-12x+36$[/tex]?

A. [tex]$x=-6$[/tex] and [tex][tex]$x=6$[/tex][/tex]
B. [tex]$x=6$[/tex] only
C. [tex]$x=-4$[/tex] and [tex][tex]$x=9$[/tex][/tex]
D. [tex]$x=-6$[/tex] only



Answer :

To determine the zeros of the quadratic function [tex]\( f(x) = x^2 - 12x + 36 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex].

Given the quadratic equation:
[tex]\[ x^2 - 12x + 36 = 0 \][/tex]

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use factoring here for simplicity.

### Step-by-Step Solution

1. Rewrite the quadratic equation in standard form:
[tex]\[ x^2 - 12x + 36 = 0 \][/tex]

2. Identify possible factors:
Look for two numbers that multiply to the constant term [tex]\( 36 \)[/tex] and add up to the coefficient of the linear term, [tex]\( -12 \)[/tex].

3. Factor the quadratic expression:
Notice that the quadratic can be factored as follows:
[tex]\[ x^2 - 12x + 36 = (x - 6)(x - 6) = (x - 6)^2 \][/tex]

4. Set each factor equal to zero:
Since this is a perfect square, we only have one unique factor:
[tex]\[ (x - 6)^2 = 0 \][/tex]

5. Solve for [tex]\( x \)[/tex]:
[tex]\[ x - 6 = 0 \][/tex]
[tex]\[ x = 6 \][/tex]

### Conclusion
The quadratic equation [tex]\( x^2 - 12x + 36 = 0 \)[/tex] has a single zero, [tex]\( x = 6 \)[/tex]. Therefore, the correct answer is:

[tex]\[ B. \ x = 6 \ \text{only} \][/tex]

So, the zeros of the function [tex]\( f(x) = x^2 - 12x + 36 \)[/tex] are:
- [tex]\( x = 6 \)[/tex] only

Hence, the correct answer is B.