If `x + y=13 and x ^(2) + y ^(2) = 89,` then find xy is

Options

A. 80
B. 169
C. 258
D. 40

Answer

To solve the problem, we will use the given equations and the identity related to squares.

**Step 1: Write down the given equations.**
We have:
1. \( x + y = 13 \)
2. \( x^2 + y^2 = 89 \)

**Step 2: Use the identity for the square of a sum.**
We know the identity:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
We can rearrange this to find \( 2xy \):
\[ 2xy = (x + y)^2 - (x^2 + y^2) \]

**Step 3: Substitute the known values into the identity.**
Substituting \( x + y = 13 \) and \( x^2 + y^2 = 89 \):
\[ 2xy = (13)^2 - 89 \]

**Step 4: Calculate \( (13)^2 \).**
Calculating \( (13)^2 \):
\[ (13)^2 = 169 \]

**Step 5: Substitute this value back into the equation for \( 2xy \).**
Now substituting back:
\[ 2xy = 169 - 89 \]

**Step 6: Perform the subtraction.**
Calculating \( 169 - 89 \):
\[ 169 - 89 = 80 \]

**Step 7: Solve for \( xy \).**
Now we have:
\[ 2xy = 80 \]
To find \( xy \), we divide both sides by 2:
\[ xy = \frac{80}{2} = 40 \]

Thus, the value of \( xy \) is \( 40 \).

**Final Answer:**
\( xy = 40 \)

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