Let's break this down step by step and solve for $X$.

The equation is:

$\left( (306 \times X) \times (260 \times X) + \pi \times (71 \times X) \times (153 \times X) \right) \times 2 + \left( 70.7530918052 \times X \right)^2 = 466.6667$

Step 1: Simplify the terms inside the parentheses

First, let's simplify the terms in the parentheses.

1. $(306 \times X) \times (260 \times X)$: $(306 \times X) \times (260 \times X) = 306 \times 260 \times X^2 = 79,560 \times X^2$

2. $\pi \times (71 \times X) \times (153 \times X)$: $\pi \times (71 \times X) \times (153 \times X) = \pi \times 71 \times 153 \times X^2 = \pi \times 10,863 \times X^2$ Using $\pi \approx 3.1416$: $\pi \times 10,863 \times X^2 \approx 34,139.5 \times X^2$

Step 2: Rewrite the equation

Now substitute these simplified expressions back into the original equation:

$\left( 79,560 \times X^2 + 34,139.5 \times X^2 \right) \times 2 + (70.7530918052 \times X)^2 = 466.6667$

This becomes:

$\left( 113,699.5 \times X^2 \right) \times 2 + (70.7530918052 \times X)^2 = 466.6667$

Now, simplify further:

$227,399 \times X^2 + (70.7530918052 \times X)^2 = 466.6667$

Step 3: Expand and simplify the second squared term

Next, expand $(70.7530918052 \times X)^2$:

$(70.7530918052 \times X)^2 = 70.7530918052^2 \times X^2$ $70.7530918052^2 \approx 5,002.85$ So: $(70.7530918052 \times X)^2 \approx 5,002.85 \times X^2$

Step 4: Final simplified equation

Substitute this back into the equation:

$227,399 \times X^2 + 5,002.85 \times X^2 = 466.6667$

Now combine the $X^2$ terms:

$(227,399 + 5,002.85) \times X^2 = 466.6667$ $232,401.85 \times X^2 = 466.6667$

Step 5: Solve for $X^2$

To isolate $X^2$, divide both sides of the equation by 232,401.85:

$X^2 = \frac{466.6667}{232,401.85}$ $X^2 \approx 0.002008$

Step 6: Solve for $X$

Finally, take the square root of both sides:

$X \approx \sqrt{0.002008} \approx 0.04487$

Final Answer:

$X \approx 0.04487$

Would you like further clarification or additional details on any of the steps?

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Follow-up Questions:

1. How would the solution change if a different value of $\pi$ is used?
2. What happens if the equation is modified by adding a constant term to the left side?
3. How can we simplify large expressions involving both multiplication and addition of variables?
4. What is the significance of having both linear and squared terms in an equation?
5. Can you think of any real-world situations where equations like this might arise?

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Tip: When solving equations with both linear and quadratic terms, try to break down the equation into smaller, manageable steps to avoid mistakes.