Solve the equation for all values of x.

2x(7x+1)(9x^2-25)=0

To solve the equation \( 2x(7x + 1)(9x^2 - 25) = 0 \), we apply the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

The equation can be broken down into three factors:

\[
2x = 0, \quad (7x + 1) = 0, \quad (9x^2 - 25) = 0
\]

### 1. Solve \( 2x = 0 \):
\[
x = 0
\]

### 2. Solve \( 7x + 1 = 0 \):
\[
7x = -1 \quad \Rightarrow \quad x = -\frac{1}{7}
\]

### 3. Solve \( 9x^2 - 25 = 0 \):
This is a difference of squares, which can be factored as:
\[
(3x - 5)(3x + 5) = 0
\]

Setting each factor equal to zero:
\[
3x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{3}
\]
\[
3x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{3}
\]

### Final Solution:
The solutions to the equation are:
\[
x = 0, \quad x = -\frac{1}{7}, \quad x = \frac{5}{3}, \quad x = -\frac{5}{3}
\]

Would you like further details on any of these steps?

Here are five related questions to consider:
1. How do you factor a quadratic equation like \(9x^2 - 25\)?
2. What is the zero-product property, and how is it applied in solving equations?
3. How can you check if your solutions to a polynomial equation are correct?
4. What happens if there are more terms in the equation? How do you handle them?
5. How do difference of squares help simplify equations?

**Tip**: Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the equation 2x(7x + 1)(9x^2 - 25) = 0 using the zero-product property and factoring techniques. This step-by-step solution covers algebraic concepts such as the difference of squares and linear equations. Ideal for students in Grades 8-10.

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