If x and y are positive integers and `sqrt(x) = y + 3` ,then what is the value of `y^(2)`?

Options

A. `x-9`
B. `x+9`
C. `x-6sqrt(x)+9`
D. `x^(2)-6sqrt(x)+9`

Answer

To solve the problem, we start with the equation given in the question:

1. **Given Equation**:
\[ \sqrt{x} = y + 3 \]

2. **Rearranging the Equation**:
We want to express \(y\) in terms of \(x\). We can rearrange the equation by moving 3 to the other side:
\[ \sqrt{x} - 3 = y \]

3. **Squaring Both Sides**:
Now, we will square both sides of the equation to eliminate the square root:
\[ (\sqrt{x} - 3)^2 = y^2 \]

4. **Applying the Identity**:
We will use the identity \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = \sqrt{x}\) and \(b = 3\):
\[ (\sqrt{x})^2 - 2 \cdot \sqrt{x} \cdot 3 + 3^2 = y^2 \]

5. **Simplifying the Expression**:
Now we simplify the left-hand side:
\[ x - 6\sqrt{x} + 9 = y^2 \]

6. **Final Expression for \(y^2\)**:
Thus, we have:
\[ y^2 = x - 6\sqrt{x} + 9 \]

Now, we have found the value of \(y^2\) in terms of \(x\):
\[ y^2 = x - 6\sqrt{x} + 9 \]