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.
