Question: User: 10x squared -14x -12=0 find zeros using zero product property
1. **State the problem:** Find the zeros of the quadratic equation $$10x^2 - 14x - 12 = 0$$ 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:**
First, find two numbers that multiply to $$10 \times (-12) = -120$$ and add to $$-14$$.
These numbers are $$-20$$ and $$6$$ because $$-20 \times 6 = -120$$ and $$-20 + 6 = -14$$.
Rewrite the middle term:
$$10x^2 - 20x + 6x - 12 = 0$$
Group terms:
$$ (10x^2 - 20x) + (6x - 12) = 0$$
Factor each group:
$$10x(x - 2) + 6(x - 2) = 0$$
Factor out the common binomial:
$$(10x + 6)(x - 2) = 0$$
4. **Simplify the first factor:**
$$10x + 6 = 2(5x + 3)$$, so the equation becomes:
$$2(5x + 3)(x - 2) = 0$$
Since $$2 \neq 0$$, we focus on:
$$(5x + 3)(x - 2) = 0$$
5. **Apply zero product property:**
Set each factor equal to zero:
$$5x + 3 = 0$$ or $$x - 2 = 0$$
6. **Solve each equation:**
For $$5x + 3 = 0$$:
$$5x = -3$$
$$x = \frac{-3}{5}$$
For $$x - 2 = 0$$:
$$x = 2$$
7. **Final answer:**
The zeros of the quadratic equation are $$x = \frac{-3}{5}$$ and $$x = 2$$.