Factoring the Quadratic: Find the Product Representation of f(x) = x² - 3x - 18

To solve the problem of factoring the quadratic expression $f(x) = x^2 - 3x - 18$, we will use the following method:

• Step 1: Identify and understand the quadratic expression, which is given in standard form: $ax^2 + bx + c$. For this expression, $a = 1$, $b = -3$, and $c = -18$.
• Step 2: Compute the product of $a$ and $c$, which yields $1 \cdot (-18) = -18$. We need to find two numbers whose product is $-18$ and whose sum is $-3$.
• Step 3: Look for pairs of factors of $-18$: - $1, -18$ - $-1, 18$ - $2, -9$ - $-2, 9$ - $3, -6$ - $-3, 6$

Among these, the pair $(3, -6)$ adds up to $-3$ and multiplies to $-18$.

• Step 4: Rewrite the quadratic expression using these numbers to represent the middle term:
  $x^2 - 3x - 18 = x^2 + 3x - 6x - 18$.
• Step 5: Group the terms to facilitate factoring:
  $(x^2 + 3x) + (-6x - 18)$.
• Step 6: Factor out the common factors in each grouped terms:
  $x(x + 3) - 6(x + 3)$.
• Step 7: Factor out the common binomial:
  $(x - 6)(x + 3)$.

Therefore, the factorized form of the quadratic function $f(x) = x^2 - 3x - 18$ is $(x - 6)(x + 3)$.