The expression $x^2 - 3x - 18$ is a quadratic equation. To better analyze or solve it, let’s consider factoring or other methods.

Step 1: Write the equation

$x^2 - 3x - 18 = 0$

Step 2: Find factors of $-18$ that add up to $-3$

The factors of $-18$ are pairs of numbers whose product is $-18$: $(1, -18), (-1, 18), (2, -9), (-2, 9), (3, -6), (-3, 6)$ The pair that sums to $-3$ is $(3, -6)$.

Step 3: Factor the quadratic

Using the pair $(3, -6)$, rewrite the middle term $-3x$ as $3x - 6x$: $x^2 - 6x + 3x - 18 = 0$

Group terms: $(x^2 - 6x) + (3x - 18) = 0$

Factor each group: $x(x - 6) + 3(x - 6) = 0$

Factor out the common term $(x - 6)$: $(x - 6)(x + 3) = 0$

Step 4: Solve for $x$

Set each factor equal to zero: $x - 6 = 0 \quad \text{or} \quad x + 3 = 0$

Solve: $x = 6 \quad \text{or} \quad x = -3$

Final Answer:

The solutions are: $x = 6 \quad \text{and} \quad x = -3$

---

Would you like details about graphing, verifying, or another method?

Related Questions:

1. How do you factor other quadratic equations like $x^2 + 5x + 6$?
2. What is the quadratic formula, and how is it used?
3. Can all quadratic equations be factored neatly like this one?
4. How can the discriminant confirm the number of solutions?
5. What are the applications of solving quadratic equations?

Tip:

Always check your solutions by substituting them back into the original equation to ensure they satisfy it.