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To solve the equation \( 2\sqrt{x} - 14 = \frac{288}{\sqrt{x}} \), let's follow these steps:

1. Multiply both sides of the equation by \(\sqrt{x}\) to eliminate the fraction:
\[ 2x - 14\sqrt{x} = 288 \]

2. Rearrange the equation to form a quadratic equation in terms of \(\sqrt{x}\):
\[ 2x - 14\sqrt{x} - 288 = 0 \]

Let \( u = \sqrt{x} \). Therefore, \( u^2 = x \). Substituting \( u \) into the equation:
\[ 2u^2 - 14u - 288 = 0 \]

3. Solve the quadratic equation \( 2u^2 - 14u - 288 = 0 \):
\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 2 \), \( b = -14 \), and \( c = -288 \).

First, calculate the discriminant:
\[ b^2 - 4ac = (-14)^2 - 4 \cdot 2 \cdot (-288) \]
\[ = 196 + 2304 \]
\[ = 2500 \]

Now, solve for \( u \):
\[ u = \frac{14 \pm \sqrt{2500}}{4} \]
\[ = \frac{14 \pm 50}{4} \]

This gives us two solutions:
\[ u = \frac{14 + 50}{4} = \frac{64}{4} = 16 \]
\[ u = \frac{14 - 50}{4} = \frac{-36}{4} = -9 \]

Since \( u = \sqrt{x} \), and the square root must be non-negative, we discard \( u = -9 \).

Therefore, \( \sqrt{x} = 16 \), which means:
\[ x = 16^2 = 256 \]

So, the solution is:
\[ x = 256 \]

Would you like more details or have any questions?

Here are some related questions you might ask:
1. Can you show more examples of solving similar equations?
2. How do you solve quadratic equations with different coefficients?
3. What are some common mistakes to avoid when solving quadratic equations?
4. How can you verify the solution of a quadratic equation?
5. What if the discriminant is negative in a quadratic equation?

**Tip:** Always check the solutions of your quadratic equations by substituting them back into the original equation to ensure they satisfy it.

Learn how to solve the equation 2√(x) - 14 = 288/√(x) step-by-step with detailed explanations and calculations. This problem involves quadratic equations and roots of equations, suitable for advanced high school students.

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