The sum of two numbers is 4 and their product is 3. Find : (i) the sum of their squares.

Answer

To solve the problem step by step, we will follow the given information and use the appropriate mathematical identities.

### Step 1: Define the variables
Let the first number be \( x \) and the second number be \( y \).

### Step 2: Set up the equations
From the problem, we know:
1. The sum of the two numbers:
\[ x + y = 4 \]
2. The product of the two numbers:
\[ x \cdot y = 3 \]

### Step 3: Use the identity for the sum of squares
We need to find the sum of their squares, which is \( x^2 + y^2 \). We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]

### Step 4: Substitute the known values
Now, we substitute the values we have into the identity:
- We know \( x + y = 4 \), so \( (x + y)^2 = 4^2 = 16 \).
- We also know \( xy = 3 \), so \( 2xy = 2 \cdot 3 = 6 \).

### Step 5: Calculate \( x^2 + y^2 \)
Now we can substitute these values into the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy = 16 - 6 = 10 \]

### Final Answer
Thus, the sum of the squares of the two numbers is:
\[ x^2 + y^2 = 10 \]

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