If `xy = 56 and x ^(2) + y ^(2) = 113,` then what will be the values of `(x+y)` ?

Options

A. 29
B. 21
C. 36
D. 15

Answer

To find the value of \( x + y \) given the equations \( xy = 56 \) and \( x^2 + y^2 = 113 \), we can use the identity:

\[ (x + y)^2 = x^2 + y^2 + 2xy \]

### Step 1: Substitute the known values into the identity

We know:
- \( x^2 + y^2 = 113 \)
- \( xy = 56 \)

Substituting these values into the identity:

\[ (x + y)^2 = 113 + 2(56) \]

### Step 2: Calculate \( 2xy \)

Now, calculate \( 2xy \):

\[ 2xy = 2 \times 56 = 112 \]

### Step 3: Add the values

Now substitute \( 112 \) back into the equation:

\[ (x + y)^2 = 113 + 112 \]

\[ (x + y)^2 = 225 \]

### Step 4: Take the square root

To find \( x + y \), we take the square root of both sides:

\[ x + y = \sqrt{225} \]

\[ x + y = 15 \]

### Final Answer

Thus, the value of \( x + y \) is \( 15 \).

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