Question: 2 — Basic Quadratic Factoring
LEARNING OBJECTIVE: Factor a quadratic expression in the form $x^2+bx+c$.
The factored form of the quadratic $x^2-4x-12$ is ________.
a.) $(x +2)(x -6)$
b.) $(x +3)(x -4)$
c.) $(x -3)(x -4)$
d.) $(x -2)(x -6)$
1. **State the problem:** Factor the quadratic expression $$x^2 - 4x - 12$$.
2. **Recall the factoring formula:** For a quadratic in the form $$x^2 + bx + c$$, we look for two numbers that multiply to $$c$$ and add to $$b$$.
3. **Identify coefficients:** Here, $$b = -4$$ and $$c = -12$$.
4. **Find two numbers:** We need two numbers whose product is $$-12$$ and sum is $$-4$$.
5. **List factor pairs of $$-12$$:**
- $$1$$ and $$-12$$ (sum = $$-11$$)
- $$-1$$ and $$12$$ (sum = $$11$$)
- $$2$$ and $$-6$$ (sum = $$-4$$)
- $$-2$$ and $$6$$ (sum = $$4$$)
6. **Choose the correct pair:** $$2$$ and $$-6$$ multiply to $$-12$$ and add to $$-4$$.
7. **Write the factored form:** $$ (x + 2)(x - 6) $$.
8. **Verify by expansion:**
$$ (x + 2)(x - 6) = x^2 - 6x + 2x - 12 = x^2 - 4x - 12 $$.
**Final answer:** The factored form is $$(x + 2)(x - 6)$$, which corresponds to option (a).