for quadratic function f, the solutions to the equation f(x) = 0 are x = 8 and x = -4. which function could represent f. Answer using these choices: a. f(x) = x^2 + 4x - 32

Question

b. f(x) = x^2 + 12x + 32
c. f(x) = x^2 - 4x - 32
d. f(x) = x^2 - 12x + 32

Answer 1

Given that the solutions to the equation $f(x)=0$ are $x=8$ and $x=-4$, we can start by expressing the quadratic function in its factored form. The factored form is given by:

$$f(x)=a(x-r_1)(x-r_2)$$

where $r_1$ and $r_2$ are the roots. In this case, $r_1=8$ and $r_2=-4$. Thus, we can express the function as:

$$f(x)=a(x-8)(x+4)$$

To find a specific quadratic function, we can expand this:

$$f(x)=a[(x-8)(x+4)]=a[x^2+4x-8x-32]=a[x^2-4x-32]$$

Now, we can compare the general form $f(x)=a{x}^2-4ax-32a$ to each of the choices provided. We see that:

1. In $f(x)=x^2+4x-32$:

   • $a=1$, $b=4$ (not matching since $b=-4a$)

2. In $f(x)=x^2+12x+32$:

   • $a=1$, $b=12$ (not matching)

3. In $f(x)=x^2-4x-32$:

   • $a=1$, $b=-4$ (matches since $b=-4a$)

4. In $f(x)=x^2-12x+32$:

   • $a=1$, $b=-12$ (not matching)

The correct choice that matches our derived quadratic function is:

$$\boxed{{\displaystyle c.,f(x)=x^2-4x-32}}$$