The sum of two numbers is 9 and their product is 20. Find the sum of their:
squares

Answer

To solve the problem, we need to find the sum of the squares of two numbers, given that their sum is 9 and their product is 20.

### Step-by-Step Solution:

1. **Let the two numbers be \( x \) and \( y \)**.
- We know from the problem statement that:
\[ x + y = 9 \quad \text{(Equation 1)} \]
\[ xy = 20 \quad \text{(Equation 2)} \]

2. **Express \( y \) in terms of \( x \)** using Equation 1:
\[ y = 9 - x \]

3. **Substitute \( y \) in Equation 2**:
\[ x(9 - x) = 20 \]
Expanding this gives:
\[ 9x - x^2 = 20 \]

4. **Rearranging the equation**:
\[ -x^2 + 9x - 20 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ x^2 - 9x + 20 = 0 \]

5. **Factoring the quadratic equation**:
\[ (x - 5)(x - 4) = 0 \]
Setting each factor to zero gives:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

6. **Finding the corresponding values of \( y \)**:
- If \( x = 5 \):
\[ y = 9 - 5 = 4 \]
- If \( x = 4 \):
\[ y = 9 - 4 = 5 \]
- Thus, the two numbers are \( 4 \) and \( 5 \).

7. **Calculating the sum of their squares**:
\[ x^2 + y^2 = 4^2 + 5^2 \]
\[ = 16 + 25 \]
\[ = 41 \]

### Final Answer:
The sum of the squares of the two numbers is \( 41 \).