Solve each equation by factoring. $$ 35 \cdot(n+5)(2 n-3)=0 $$

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UpStudy AI · Accepted Answer

To solve the equation by factoring:

$$
35 \cdot (n + 5)(2n - 3) = 0
$$

We use the **Zero Product Property**, which states that if the product of several factors is zero, at least one of the factors must be zero.

Let's break down the equation:

1. **Factor Analysis:**
   
   The equation is already factored into three parts:
   
   - **35:** This is a constant multiplier. Since $$35 \neq 0$$, it doesn't contribute to the solution.
   - **$$n + 5$$:** Setting this factor to zero gives one possible solution.
   - **$$2n - 3$$:** Setting this factor to zero gives another possible solution.

2. **Set Each Factor to Zero:**

- **First Factor:**
     $$
     n + 5 = 0
     $$
     $$
     n = -5
     $$
   
   - **Second Factor:**
     $$
     2n - 3 = 0
     $$
     $$
     2n = 3
     $$
     $$
     n = \frac{3}{2}
     $$

3. **Solutions:**
   
   The solutions to the equation are:
   
   $$
   n = -5 \quad \text{and} \quad n = \frac{3}{2}
   $$

**Final Answer:**

The solutions are $$ n = -5 $$ and $$ n = \dfrac{3}{2} $$.
The solutions are $$ n = -5 $$ and $$ n = \frac{3}{2} $$.

Solve each equation by factoring. \[ 35 \cdot(n+5)(2 n-3)=0 \]

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