If `(x-y)^(2)=50 and xy=7`, what is the value of `x^(2)+y^(2)`?

Options

A. `8`
B. `36`
C. `43`
D. `64`

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) given the equations:

1. \( (x - y)^2 = 50 \)
2. \( xy = 7 \)

### Step-by-Step Solution:

**Step 1: Rewrite the first equation using the identity.**

We know that:
\[ (x - y)^2 = x^2 + y^2 - 2xy \]
Substituting the first equation into this identity gives us:
\[ 50 = x^2 + y^2 - 2xy \]

**Step 2: Substitute the value of \( xy \).**

We have \( xy = 7 \). Therefore, \( 2xy = 2 \times 7 = 14 \). Substituting this into the equation:
\[ 50 = x^2 + y^2 - 14 \]

**Step 3: Rearrange the equation to isolate \( x^2 + y^2 \).**

Now, we can add 14 to both sides of the equation:
\[ 50 + 14 = x^2 + y^2 \]
This simplifies to:
\[ 64 = x^2 + y^2 \]

**Step 4: Conclusion.**

Thus, the value of \( x^2 + y^2 \) is:
\[ \boxed{64} \]