Solving the Square Root Equation: Find (x, y) for √x - √y = √(√61 - 6) and xy = 9

To solve the problem, we will proceed with the following steps:

• Step 1: Calculate the value of $\sqrt{\sqrt{61}-6}$.
• Step 2: Express $\sqrt{y}$ in terms of $\sqrt{x}$ using the first equation.
• Step 3: Form a single-variable equation to solve for $\sqrt{x}$.
• Step 4: Back-substitute to find $\sqrt{y}$.
• Step 5: Use squaring to find $x$ and $y$ as needed.

Step 1: Compute $\sqrt{\sqrt{61}-6}$.

Calculate $\sqrt{61}-6 \to \sqrt{61} \approx 7.81$. Therefore, $\sqrt{61}-6 \approx 1.81$. Thus $\sqrt{\sqrt{61}-6} = \sqrt{1.81}$. For efficacy, we solve further using variables.

Step 2: Using the equation $\sqrt{x} - \sqrt{y} = \sqrt{\sqrt{61}-6}$, let $\sqrt{x} = a$ and $\sqrt{y} = b$ with $a-b = c$ and referred c as calculated.

Step 3: With $ab = \sqrt{9} = 3$ (as $xy = 9$ hence $\sqrt{x}\sqrt{y}$), we substitute $b = \frac{3}{a}$.

Thus, $a - \frac{3}{a} = \sqrt{\sqrt{61} - 6}$. Rearrange into: $a^2 - a\sqrt{\sqrt{61} - 6} - 3 = 0$ as a quadratic equation in $a$.

Solving yields solutions for $a$, use quadratic formula, or completing squares.

Solving, get solutions, $a = \frac{\sqrt{61}}{2} - 2.5$ and $\frac{\sqrt{61}}{2} + 2.5$

Backward solve $b$ by substituting values back.

Thus, for each $a$, solve for $x$ or $y$ square them and check.

The solution is:

$x=\frac{\sqrt{61}}{2}-2.5$, $y=\frac{\sqrt{61}}{2}+2.5$ or $x=\frac{\sqrt{61}}{2}+2.5$, $y=\frac{\sqrt{61}}{2}-2.5$

Final solution:

$x=\frac{\sqrt{61}}{2}-2.5$

$y=\frac{\sqrt{61}}{2}+2.5$

or

$x=\frac{\sqrt{61}}{2}+2.5$

$y=\frac{\sqrt{61}}{2}-2.5$