If x-y=5 , xy=24 then the value of `x^2+y^2` will be

Options

A. 23
B. 73
C. 65
D. 74

Answer

To find the value of \( x^2 + y^2 \) given the equations \( x - y = 5 \) and \( xy = 24 \), we can follow these steps:

### Step 1: Square the first equation
We start with the equation:
\[ x - y = 5 \]
Squaring both sides gives us:
\[ (x - y)^2 = 5^2 \]
This expands to:
\[ x^2 - 2xy + y^2 = 25 \]

### Step 2: Substitute the value of \( xy \)
We know from the problem that:
\[ xy = 24 \]
Now, we can substitute \( xy \) into the equation we obtained from squaring:
\[ x^2 - 2(24) + y^2 = 25 \]
This simplifies to:
\[ x^2 - 48 + y^2 = 25 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + y^2 \):
\[ x^2 + y^2 = 25 + 48 \]
This simplifies to:
\[ x^2 + y^2 = 73 \]

### Final Answer
Thus, the value of \( x^2 + y^2 \) is:
\[ \boxed{73} \]