Factorize x2 + 9x + 20

Analysis

To factorize the quadratic expression $x^2 + 9x + 20$, we need to find two numbers that satisfy two conditions:

1. Their product must be equal to the constant term ($20$).
2. Their sum must be equal to the coefficient of the middle term ($9$).

Concept: The AC Method (Factorization of Quadratics)

For a quadratic expression in the form $ax^2 + bx + c$, we look for two factors of $ac$ that add up to $b$. In this case, $a=1$, $b=9$, and $c=20$. We need two numbers that multiply to give $20$ and add to give $9$.

Step-by-Step Solution

1. List the factors of 20: • $1 \times 20$ (Sum = $21$) • $2 \times 10$ (Sum = $12$) • $4 \times 5$ (Sum = $9$)

2. Select the correct pair: The pair $4$ and $5$ satisfies both conditions (product is $20$, sum is $9$).

3. Rewrite the middle term: Replace $9x$ with $4x + 5x$: $x^2 + 4x + 5x + 20$

4. Factor by grouping: Group the first two terms and the last two terms: $(x^2 + 4x) + (5x + 20)$

5. Factor out the common terms: $x(x + 4) + 5(x + 4)$

6. Final grouping: Since $(x + 4)$ is common to both terms, we factor it out: $(x + 5)(x + 4)$

Conclusion

Comparing our result to the given options, the correct factorization is $(x + 5)(x + 4)$.

Exam Tip: When the constant term ($c$) is positive, the two numbers you are looking for must have the same sign (both positive or both negative). Since the middle term ($9x$) is positive, you know immediately that both numbers must be positive!