Question: 6x squared minus 7x minus 20 =0 find the zeros using zero product property
1. **State the problem:** We need to find the zeros of the quadratic equation $$6x^2 - 7x - 20 = 0$$ using the zero product property.
2. **Recall the zero product property:** If $$ab = 0$$, then either $$a = 0$$ or $$b = 0$$.
3. **Factor the quadratic:** We look for two numbers that multiply to $$6 \times (-20) = -120$$ and add to $$-7$$.
4. The numbers are $$8$$ and $$-15$$ because $$8 \times (-15) = -120$$ and $$8 + (-15) = -7$$.
5. Rewrite the middle term:
$$6x^2 + 8x - 15x - 20 = 0$$
6. Group terms:
$$(6x^2 + 8x) + (-15x - 20) = 0$$
7. Factor each group:
$$2x(3x + 4) - 5(3x + 4) = 0$$
8. Factor out the common binomial:
$$(3x + 4)(2x - 5) = 0$$
9. Apply zero product property:
$$3x + 4 = 0 \quad \text{or} \quad 2x - 5 = 0$$
10. Solve each equation:
$$3x = -4 \Rightarrow x = \frac{-4}{3}$$
$$2x = 5 \Rightarrow x = \frac{5}{2}$$
**Final answer:** The zeros are $$x = \frac{-4}{3}$$ and $$x = \frac{5}{2}$$.