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:
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In \(f(x) = x^2 + 4x - 32\):
- \(a = 1\), \(b = 4\) (not matching since \(b = -4a\))
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In \(f(x) = x^2 + 12x + 32\):
- \(a = 1\), \(b = 12\) (not matching)
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In \(f(x) = x^2 - 4x - 32\):
- \(a = 1\), \(b = -4\) (matches since \(b = -4a\))
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In \(f(x) = x^2 - 12x + 32\):
- \(a = 1\), \(b = -12\) (not matching)
The correct choice that matches our derived quadratic function is:
\[ \boxed{c. , f(x) = x^2 - 4x - 32} \]
3 answers