The sum of two numbers is 59 and their product is 840. Find the sum of their squares.

Options

A. 2961
B. 1801
C. 1875
D. 1754

Answer

To solve the problem step by step, we will follow the mathematical principles outlined in the video transcript.

### Step 1: Define the Variables
Let the two numbers be \( X \) and \( Y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the two numbers:
\[ X + Y = 59 \quad \text{(Equation 1)} \]
2. The product of the two numbers:
\[ XY = 840 \quad \text{(Equation 2)} \]

### Step 3: Use the Formula for the Sum of Squares
We want 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
Now, substitute the values from Equation 1 and Equation 2 into the formula:
\[ X^2 + Y^2 = (59)^2 - 2(840) \]

### Step 5: Calculate \( (X + Y)^2 \)
Calculate \( 59^2 \):
\[ 59^2 = 3481 \]

### Step 6: Calculate \( 2XY \)
Calculate \( 2 \times 840 \):
\[ 2 \times 840 = 1680 \]

### Step 7: Substitute and Simplify
Now substitute these values back into the equation:
\[ X^2 + Y^2 = 3481 - 1680 \]

### Step 8: Final Calculation
Now perform the subtraction:
\[ X^2 + Y^2 = 3481 - 1680 = 1801 \]

### Conclusion
Thus, the sum of the squares of the two numbers is:
\[ \boxed{1801} \]