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 \(f(x) = ax^2 - 4ax - 32a\) 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:

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