Factorize $x^2 − 2x − 15$

Analysis

To factorize the quadratic expression $x^2 - 2x - 15$, we need to find two numbers that satisfy two conditions:

1. Their product must be equal to the constant term ($-15$).
2. Their sum must be equal to the coefficient of the middle term ($-2$).

Concept: Quadratic Factorization

A quadratic expression in the form $ax^2 + bx + c$ can be factorized by splitting the middle term ($bx$). We look for two factors of $ac$ that add up to $b$. Once found, we rewrite the expression and factor by grouping.

Step-by-Step Solution

Step 1: Find the two numbers. We need two numbers that multiply to $-15$ and add to $-2$.

• Factors of $-15$ include: • $1$ and $-15$ (Sum = $-14$) • $-1$ and $15$ (Sum = $14$) • $3$ and $-5$ (Sum = $-2$) <-- This is our pair!

Step 2: Rewrite the middle term. Replace $-2x$ with $+3x - 5x$: $x^2 + 3x - 5x - 15$

Step 3: Factor by grouping. Group the first two terms and the last two terms: $(x^2 + 3x) + (-5x - 15)$

Factor out the common terms from each group: $x(x + 3) - 5(x + 3)$

Step 4: Final factorization. Since $(x + 3)$ is common to both terms, we factor it out: $(x - 5)(x + 3)$

Conclusion

Comparing our result $(x - 5)(x + 3)$ with the given options, it matches option D.

Exam Tip

When factorizing, always check your signs carefully! If the constant term is negative (like $-15$), one of your two numbers must be positive and the other must be negative. If the middle term is negative, the larger of the two numbers must carry the negative sign.