To solve the quadratic equation $3x^2 - 17x - 6 = 0$ by factorizing, follow these steps:

Step 1: Multiply the coefficient of $x^2$ (which is 3) by the constant term (which is -6).

$3 \times -6 = -18$

Step 2: Find two numbers that multiply to $-18$ and add up to the coefficient of $x$ (which is -17).

The numbers that work are $-18$ and $1$, because: $-18 \times 1 = -18 \quad \text{and} \quad -18 + 1 = -17$

Step 3: Rewrite the middle term $-17x$ using these two numbers:

$3x^2 - 18x + 1x - 6 = 0$

Step 4: Factor by grouping:

Group the terms into two pairs: $(3x^2 - 18x) + (1x - 6) = 0$

Factor out the greatest common factor (GCF) from each pair: $3x(x - 6) + 1(x - 6) = 0$

Step 5: Factor out the common binomial factor:

$(3x + 1)(x - 6) = 0$

Step 6: Set each factor equal to zero and solve for $x$:

$3x + 1 = 0 \quad \text{or} \quad x - 6 = 0$

For $3x + 1 = 0$: $3x = -1 \quad \Rightarrow \quad x = -\frac{1}{3}$

For $x - 6 = 0$: $x = 6$

Final Answer:

The solutions are: $x = 6 \quad \text{or} \quad x = -\frac{1}{3}$

Would you like more details on any of the steps or have any questions?

Here are five related questions to expand on this topic:

1. How can you check if the factorization is correct?
2. When is factorization more convenient than the quadratic formula?
3. How do you factorize when the quadratic equation doesn’t easily factorize?
4. Can all quadratic equations be solved by factorization?
5. What are the advantages of factorization over completing the square?

Tip: Always check the factorization by multiplying the factors back out to ensure that you get the original quadratic equation.