Find all real solutions of the equation. (Enter your answers as a comma-separated list.)
6x5 − 54x = 0

We are given the equation:

\[
6x^5 - 54x = 0
\]

### Step 1: Factor the equation
First, factor out the common factor on both terms:

\[
6x(x^4 - 9) = 0
\]

### Step 2: Solve for \(x\)
Now, we have two factors to solve:

1. \(6x = 0\)
2. \(x^4 - 9 = 0\)

#### 1. \(6x = 0\)
Solving for \(x\):

\[
x = 0
\]

#### 2. \(x^4 - 9 = 0\)
This is a difference of squares:

\[
x^4 - 9 = (x^2 - 3)(x^2 + 3) = 0
\]

Now, solve each factor:
- \(x^2 - 3 = 0 \) gives \(x^2 = 3\), so \(x = \pm \sqrt{3}\).
- \(x^2 + 3 = 0\) gives \(x^2 = -3\), but there are no real solutions for this since the square root of a negative number is not real.

### Step 3: List all real solutions
The real solutions are \(x = 0\), \(x = \sqrt{3}\), and \(x = -\sqrt{3}\).

Thus, the real solutions are:

\[
x = 0, \sqrt{3}, -\sqrt{3}
\]

Would you like more details or have any questions about the process? Here are some related problems:

1. Solve \(4x^3 - 16x = 0\).
2. Solve \(x^6 - 64 = 0\).
3. Factor and solve \(x^4 - 16 = 0\).
4. Find the real solutions of \(2x^5 - 50x = 0\).
5. Solve \(x^2 + 1 = 0\) in terms of complex solutions.

**Tip**: When factoring polynomials, always look for common factors first, then check for special factoring patterns like differences of squares.

Find all real solutions of the equation. (Enter your answers as a comma-separated list.) 6x^5 − 54x = 0

Find the real solutions of the equation 6x^5 − 54x = 0 by factoring. This solution covers algebraic techniques, including factoring, difference of squares, and solving polynomial equations. Suitable for high school students (Grades 9-12).

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