If $$ (x+y)^{2}=36, x y=9 $$, then $$ x^{2}+y^{2}=\cdots $$ $$ \begin{array}{ll} ext { (a) } 4 & ext { (b) } 27\end{array} $$

Question

If $$ (x+y)^{2}=36, x y=9 $$, then $$ x^{2}+y^{2}=\cdots $$ $$ \begin{array}{ll}	ext { (a) } 4 & 	ext { (b) } 27\end{array} $$

UpStudy AI · Accepted Answer

To find $$ x^2 + y^2 $$ given the equations $$ (x+y)^2 = 36 $$ and $$ xy = 9 $$, we can use the identity:

$$
x^2 + y^2 = (x+y)^2 - 2xy
$$

### Step 1: Calculate $$ x+y $$

From the equation $$ (x+y)^2 = 36 $$, we can find $$ x+y $$:

$$
x+y = \sqrt{36} = 6 \quad \text{(considering the positive root)}
$$

### Step 2: Substitute into the identity

Now we can substitute $$ x+y $$ and $$ xy $$ into the identity:

$$
x^2 + y^2 = (x+y)^2 - 2xy
$$

Substituting the known values:

$$
x^2 + y^2 = 36 - 2 \cdot 9
$$

### Step 3: Calculate $$ x^2 + y^2 $$

Now, calculate:

$$
x^2 + y^2 = 36 - 18 = 18
$$

### Conclusion

The value of $$ x^2 + y^2 $$ is $$ 18 $$. However, this value is not listed in the options provided. Let's double-check the calculations:

1. $$ (x+y)^2 = 36 $$ gives $$ x+y = 6 $$.
2. $$ xy = 9 $$.
3. Using the identity: 
   $$
   x^2 + y^2 = 36 - 18 = 18
   $$

Since $$ 18 $$ is not among the options (a) 4 or (b) 27, it seems there might be a misunderstanding in the options provided.

Thus, the final answer is:

$$
\boxed{18}
$$
$$ x^2 + y^2 = 18 $$

If \( (x+y)^{2}=36, x y=9 \), then \( x^{2}+y^{2}=\cdots \) \( \begin{array}{ll}\text { (a) } 4 & \text { (b) } 27\end{array} \)