he product of two numbers is 63 . The sum of the two numbers is 16 . /hat is the difference between the two numbers? Show your thinking

To find the difference between two numbers given that their product is 63 and their sum is 16, we can follow these steps: 1. **Define the Variables:** - Let the two numbers be \( x \) and \( y \). 2. **Set Up the Equations:** - **Product Equation:** \( x \times y = 63 \) \( xy = 63 \) - **Sum Equation:** \( x + y = 16 \) 3. **Express One Variable in Terms of the Other:** - From the sum equation: \( y = 16 - x \) 4. **Substitute into the Product Equation:** - Replace \( y \) in \( xy = 63 \) with \( 16 - x \): \( x(16 - x) = 63 \) \( 16x - x^2 = 63 \) \( -x^2 + 16x - 63 = 0 \) Multiply both sides by -1 to make it easier: \( x^2 - 16x + 63 = 0 \) 5. **Solve the Quadratic Equation:** - The quadratic equation is \( x^2 - 16x + 63 = 0 \). - To solve for \( x \), we can factor the equation: \( (x - 7)(x - 9) = 0 \) So, \( x = 7 \) or \( x = 9 \). 6. **Find the Corresponding \( y \) Values:** - If \( x = 7 \), then \( y = 16 - 7 = 9 \). - If \( x = 9 \), then \( y = 16 - 9 = 7 \). 7. **Calculate the Difference:** - The two numbers are 7 and 9. - Difference = \( |9 - 7| = 2 \). **Answer:** The difference between the two numbers is **2**.