The expression you provided, as written, contains some symbols that might not be correctly formatted or clear. It seems to mix different notations like roots, powers, and possibly multiplications. Here's what I interpret it to be:

\[
\frac{2x}{8x} + x^2 = 64 \sqrt{2^3} x
\]

If this interpretation is correct, let's solve it step by step.

### Step 1: Simplify the left side

\[
\frac{2x}{8x} + x^2 = 64 \sqrt{8} x
\]

The first term simplifies as:

\[
\frac{2x}{8x} = \frac{2}{8} = \frac{1}{4}
\]

So the equation becomes:

\[
\frac{1}{4} + x^2 = 64 \sqrt{8} x
\]

### Step 2: Simplify the square root

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

Thus, the equation now becomes:

\[
\frac{1}{4} + x^2 = 64 \times 2\sqrt{2} x = 128 \sqrt{2} x
\]

### Step 3: Rearrange the equation

Now, we have:

\[
x^2 - 128 \sqrt{2} x + \frac{1}{4} = 0
\]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \).

### Step 4: Solve the quadratic equation

To solve for \(x\), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -128 \sqrt{2}\), and \(c = \frac{1}{4}\).

Let's now compute the discriminant:

\[
b^2 - 4ac = (-128 \sqrt{2})^2 - 4 \times 1 \times \frac{1}{4}
\]

Let me calculate the exact solution.The solutions to the quadratic equation are:

\[
x_1 = 181.02 \quad \text{and} \quad x_2 = 0.0014
\]

Both solutions are valid, and the discriminant confirms that there are two distinct real roots.

Would you like a more detailed explanation or have any questions about these steps?

### Related Questions:
1. How do you factor quadratic equations like this one?
2. Can you explain the importance of the discriminant in quadratic equations?
3. How does the quadratic formula work, and when is it used?
4. Can you solve quadratic equations graphically?
5. How do roots and solutions of quadratic equations relate to the graph of the function?

### Tip:
Always check the discriminant before solving a quadratic equation. It helps determine the nature of the roots—real or complex!

Solve the quadratic equation derived from simplifying 2x/8x + x^2 = 64√2³x, step by step. This solution involves simplifying expressions, using radicals, and applying the quadratic formula to find the real roots. Suitable for students in Grades 10-12.

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