To solve the equation [tex]\(3p(5p - 1) = 0\)[/tex], we follow these steps:
1. Identify when the product equals zero:
The equation [tex]\(3p(5p - 1) = 0\)[/tex] is in a factored form. For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\(p\)[/tex].
2. Set each factor to zero and solve:
- First factor: [tex]\(3p = 0\)[/tex]
[tex]\[
3p = 0 \implies p = 0
\][/tex]
- Second factor: [tex]\(5p - 1 = 0\)[/tex]
[tex]\[
5p - 1 = 0 \implies 5p = 1 \implies p = \frac{1}{5}
\][/tex]
3. List all solutions:
We found two values of [tex]\(p\)[/tex] that satisfy the equation:
[tex]\[
p = 0 \quad \text{and} \quad p = \frac{1}{5}
\][/tex]
Thus, the solutions to the equation [tex]\(3p(5p - 1) = 0\)[/tex] are [tex]\(p = 0\)[/tex] and [tex]\(p = \frac{1}{5}\)[/tex].
From the given options, the correct solutions are [tex]\( \boxed{0} \)[/tex] and [tex]\( \boxed{\frac{1}{5}} \)[/tex].
