Answer :
To solve the equation
$$
5x^3 = 405x,
$$
we first bring all the terms to one side:
$$
5x^3 - 405x = 0.
$$
Next, factor out the common factor $5x$:
$$
5x(x^2 - 81) = 0.
$$
Now, observe that $x^2 - 81$ is a difference of squares, which factors as follows:
$$
x^2 - 81 = (x - 9)(x + 9).
$$
Thus, the equation becomes:
$$
5x(x - 9)(x + 9) = 0.
$$
For the product to be zero, at least one of the factors must be zero. So we set each factor equal to zero:
1. From $5x = 0$, we get:
$$
x = 0.
$$
2. From $x - 9 = 0$, we get:
$$
x = 9.
$$
3. From $x + 9 = 0$, we get:
$$
x = -9.
$$
Thus, the solutions to the equation are
$$
x = 0,\quad x = 9,\quad x = -9.
$$
In terms of the answer selections provided, the correct answers are $9$, $0$, and $-9$.
$$
5x^3 = 405x,
$$
we first bring all the terms to one side:
$$
5x^3 - 405x = 0.
$$
Next, factor out the common factor $5x$:
$$
5x(x^2 - 81) = 0.
$$
Now, observe that $x^2 - 81$ is a difference of squares, which factors as follows:
$$
x^2 - 81 = (x - 9)(x + 9).
$$
Thus, the equation becomes:
$$
5x(x - 9)(x + 9) = 0.
$$
For the product to be zero, at least one of the factors must be zero. So we set each factor equal to zero:
1. From $5x = 0$, we get:
$$
x = 0.
$$
2. From $x - 9 = 0$, we get:
$$
x = 9.
$$
3. From $x + 9 = 0$, we get:
$$
x = -9.
$$
Thus, the solutions to the equation are
$$
x = 0,\quad x = 9,\quad x = -9.
$$
In terms of the answer selections provided, the correct answers are $9$, $0$, and $-9$.
