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To fully factorize the quadratic expression \(x^2 - x - 42\), follow these steps:

1. Identify two numbers that multiply to give \(-42\) (the constant term) and add to give \(-1\) (the coefficient of the linear term \(x\)).

   The two numbers are \(-7\) and \(6\), because:
   \[
   -7 \times 6 = -42 \quad \text{and} \quad -7 + 6 = -1
   \]

2. Now rewrite the middle term \(-x\) as \(-7x + 6x\):
   \[
   x^2 - 7x + 6x - 42
   \]

3. Factor by grouping:
   \[
   x(x - 7) + 6(x - 7)
   \]

4. Factor out the common binomial factor \((x - 7)\):
   \[
   (x - 7)(x + 6)
   \]

Thus, the fully factorized form of \(x^2 - x - 42\) is:
\[
(x - 7)(x + 6)
\]

Would you like further details or have any questions?

Here are five related questions:
1. How can we verify that our factorization is correct?
2. Can every quadratic expression be factorized using this method?
3. What are the roots of the equation \(x^2 - x - 42 = 0\)?
4. How is the method of completing the square different from factorization?
5. What role do the discriminant and its value play in factorizing quadratics?

**Tip:** Always check your factorization by expanding the factors to see if you recover the original expression.

This page provides a step-by-step explanation on how to fully factor the quadratic equation x^2 - x - 42. Learn about key concepts such as factoring quadratic equations, factoring by grouping, and applying the zero product property. Suitable for students in Grades 8-10.

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