Question: 4 — Solving Quadratic Equations
LEARNING OBJECTIVE: Solve a quadratic equation using the Zero Factor Property of Multiplication.
What are the solutions to the equation $x^2 - 11x + 24 = 0$?
a.) $x = -3$ and $x = 8$
b.) $x = -3$ and $x = -8$
c.) $x = 3$ and $x = 8$
d.) $x = 3$ and $x = -8$
1. **State the problem:** We need to find the solutions to the quadratic equation $$x^2 - 11x + 24 = 0$$.
2. **Recall the Zero Factor Property:** If a product of two factors equals zero, then at least one of the factors must be zero. That is, if $$ab = 0$$, then $$a = 0$$ or $$b = 0$$.
3. **Factor the quadratic:** We look for two numbers that multiply to $$24$$ and add to $$-11$$. These numbers are $$-3$$ and $$-8$$ because $$-3 \times -8 = 24$$ and $$-3 + -8 = -11$$.
4. **Write the factored form:**
$$
x^2 - 11x + 24 = (x - 3)(x - 8) = 0
$$
5. **Apply the Zero Factor Property:**
$$
(x - 3) = 0 \quad \text{or} \quad (x - 8) = 0
$$
6. **Solve each equation:**
$$
x - 3 = 0 \Rightarrow x = 3
$$
$$
x - 8 = 0 \Rightarrow x = 8
$$
7. **Final answer:** The solutions are $$x = 3$$ and $$x = 8$$.
Therefore, the correct choice is **c.) $x = 3$ and $x = 8$**.