Question

select the correct answer.
the zeros of a quadratic function are 6 and -4. which of these choices could be the function?
a. ( f(x) = (x - 6)(x - 4) )
b. ( f(x) = (x - 6)(x + 4) )
c. ( f(x) = (x + 6)(x - 4) )
d. ( f(x) = (x + 6)(x + 4) )

Explanation:

Step1: Recall zero - factor property

If a quadratic function has zeros at \(x = a\) and \(x = b\), then the factored form of the quadratic function is \(f(x)=(x - a)(x - b)\). This is because if \(x=a\) is a zero, then \((x - a)=0\) when \(x = a\), and similarly for \(x = b\).

Step2: Identify the factors from zeros

We are given that the zeros of the quadratic function are \(x = 6\) and \(x=-4\).

• For the zero \(x = 6\), using the zero - factor property, the corresponding factor is \((x - 6)\) (since when \(x = 6\), \(x-6=6 - 6=0\)).
• For the zero \(x=-4\), we can rewrite \(x=-4\) as \(x+4 = 0\) (by adding 4 to both sides of the equation \(x=-4\)). So the corresponding factor is \((x + 4)\) (since when \(x=-4\), \(x + 4=-4 + 4=0\)).

Step3: Write the factored form of the function

Combining these two factors, the quadratic function in factored form is \(f(x)=(x - 6)(x + 4)\).

Answer:

B. \(f(x)=(x - 6)(x + 4)\)