Question
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Answer
\( x^{2} + 7x + 12 = (x + 3)(x + 4) \)
Solution
To factorise the quadratic expression \( x^{2} + 7x + 12 \), we need to find two numbers that:
1. **Multiply** to give the constant term (**12**).
2. **Add** up to give the coefficient of the linear term (**7**).
Let's consider the pairs of factors of **12**:
- **1 and 12**: \(1 \times 12 = 12\) and \(1 + 12 = 13\) → **Not** equal to 7.
- **2 and 6**: \(2 \times 6 = 12\) and \(2 + 6 = 8\) → **Not** equal to 7.
- **3 and 4**: \(3 \times 4 = 12\) and \(3 + 4 = 7\) → **Perfect!**
So, the numbers **3** and **4** satisfy both conditions.
**Therefore, the factorised form is:**
\[
x^{2} + 7x + 12 = (x + 3)(x + 4)
\]
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Simplify this solution Extra Insights
To factorise the quadratic \( x^{2}+7x+12 \), we look for two numbers that multiply to 12 (the constant term) and add up to 7 (the coefficient of \(x\)). The numbers 3 and 4 fit the bill perfectly because \(3 \times 4 = 12\) and \(3 + 4 = 7\). Thus, we can express the quadratic as: \( (x + 3)(x + 4) \). Great job on tackling that! Now, when you encounter quadratic expressions, remember to always check if a perfect square or other easy factors could work before diving into complex methods. Small miscalculations can lead to incorrect factors, so patience and step-by-step verification are key!
