Solving the equation 2√x - 14 = 288/√x

We are tasked with solving the equation:

$$
2\sqrt{x} - 14 = \frac{288}{\sqrt{x}}
$$

Step 1: Eliminate the denominator

To eliminate the fraction, multiply both sides of the equation by $\sqrt{x}$ (assuming $\sqrt{x} \neq 0$):

$$
\sqrt{x} \left( 2\sqrt{x} - 14 \right) = \sqrt{x} \cdot \frac{288}{\sqrt{x}}
$$

Simplify both sides:

$$
2x - 14\sqrt{x} = 288
$$

Step 2: Rearrange the equation

Rearrange the equation to bring all terms to one side:

$$
2x - 14\sqrt{x} - 288 = 0
$$

Step 3: Substitute $y = \sqrt{x}$

Let $y = \sqrt{x}$. Then $y^2 = x$. Substituting these into the equation, we get:

$$
2y^2 - 14y - 288 = 0
$$

Step 4: Simplify the quadratic equation

Divide the entire equation by 2 to simplify:

$$
y^2 - 7y - 144 = 0
$$

Step 5: Solve the quadratic equation

We solve the quadratic equation $y^2 - 7y - 144 = 0$ using the quadratic formula:

$$
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Here, $a = 1$, $b = -7$, and $c = -144$. Substitute these values into the formula:

$$
y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-144)}}{2(1)}
$$

Simplify step by step:

$$
y = \frac{7 \pm \sqrt{49 + 576}}{2}
$$

$$
y = \frac{7 \pm \sqrt{625}}{2}
$$

$$
y = \frac{7 \pm 25}{2}
$$

This gives us two solutions for $y$:

$$
y = \frac{7 + 25}{2} = \frac{32}{2} = 16
$$

$$
y = \frac{7 - 25}{2} = \frac{-18}{2} = -9
$$

Step 6: Back-substitute $y = \sqrt{x}$

Recall that $y = \sqrt{x}$. Since $\sqrt{x}$ must be non-negative, we discard $y = -9$ and keep $y = 16$.

Thus:

$$
\sqrt{x} = 16
$$

Square both sides to solve for $x$:

$$
x = 16^2 = 256
$$

Step 7: Verify the solution

Substitute $x = 256$ back into the original equation to verify:

$$
2\sqrt{256} - 14 = \frac{288}{\sqrt{256}}
$$

Calculate each term:

$$
\sqrt{256} = 16
$$

$$
2\sqrt{256} = 2 \cdot 16 = 32
$$

$$
2\sqrt{256} - 14 = 32 - 14 = 18
$$

$$
\frac{288}{\sqrt{256}} = \frac{288}{16} = 18
$$

Both sides are equal, so the solution is verified.

Final Answer:

$$
\boxed{256}
$$