The sum of two numbers is 24 and their product is 143. The sum of their squares is

Options

A. 296
B. 295
C. 290
D. 228

Answer

To solve the problem step by step, we need to find the sum of the squares of two numbers given their sum and product.

### Step 1: Define the variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Write the equations based on the problem
From the problem, we know:
1. The sum of the two numbers:
\[ x + y = 24 \]
2. The product of the two numbers:
\[ x \cdot y = 143 \]

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

### Step 4: Substitute the known values into the formula
We already have \( x + y = 24 \) and \( xy = 143 \). Now we can substitute these values into the formula:
\[ x^2 + y^2 = (24)^2 - 2 \cdot 143 \]

### Step 5: Calculate \( (x + y)^2 \)
Calculating \( (24)^2 \):
\[ (24)^2 = 576 \]

### Step 6: Calculate \( 2xy \)
Calculating \( 2 \cdot 143 \):
\[ 2 \cdot 143 = 286 \]

### Step 7: Substitute back to find \( x^2 + y^2 \)
Now substituting back into the equation:
\[ x^2 + y^2 = 576 - 286 \]

### Step 8: Perform the final calculation
Calculating \( 576 - 286 \):
\[ 576 - 286 = 290 \]

### Conclusion
Thus, the sum of the squares of the two numbers is:
\[ \boxed{290} \]