Factor completely:
$3 f^2-8 f-3$
We are given the quadratic expression \(3f^2 - 8f - 3\). To factor this trinomial of the form \(Af^2 + Bf + C\), where \(A = 3\), \(B = -8\), and \(C = -3\), we need to find two factors of the product \(A \times C\) that add up to the middle coefficient \(B\).
First, we calculate the product of the lead coefficient and the constant term: \(A \times C = 3 \times (-3) = -9\). We seek two numbers, let's call them \(p\) and \(q\), such that their product is \(-9\) and their sum is the middle coefficient \(B = -8\). By inspecting the factors of \(-9\), we see that \(-9\) and \(1\) satisfy these conditions: \((-9) \times 1 = -9\) and \((-9) + 1 = -8\).
Now we rewrite the middle term \(-8f\) using the two numbers we found: \(3f^2 - 9f + f - 3\). We can now group these into two pairs to factor by grouping: \((3f^2 - 9f) + (f - 3)\). From the first group \((3f^2 - 9f)\), we can factor out the greatest common factor, \(3f\), resulting in \(3f(f - 3)\). The second group \((f - 3)\) remains as it is, or \(1(f - 3)\).
Finally, we factor out the common binomial term \((f - 3)\):
\[3f(f - 3) + 1(f - 3) = (3f + 1)(f - 3)\]
Thus, the factored form of the expression \(3f^2 - 8f - 3\) is \((3f + 1)(f - 3)\).
Factoring quadratic expressions is essential in fields like engineering and physics. For example, if you are designing a rectangular garden where the area is given by the expression \(3f^2 - 8f - 3\), factoring this into \((3f + 1)(f - 3)\) allows you to determine the algebraic expressions for the dimensions (length and width) of the garden in terms of a variable \(f\). This technique simplifies complex geometric problems and helps in finding optimal designs.