Two factors of -48 have a difference of 19. The factor with a greater absolute value is positive. What is the sum of the factors?

Set up the equations based on the given information: x × y = − 48 and x − y = 19 .
Express x in terms of y : x = y + 19 .
Substitute into the first equation and rearrange to form a quadratic equation: y 2 + 19 y + 48 = 0 .
Solve the quadratic equation by factoring: ( y + 16 ) ( y + 3 ) = 0 , so y = − 16 or y = − 3 . Find corresponding x values: x = 3 or x = 16 .
Choose the correct pair of factors where the factor with greater absolute value is positive: x = 16 and y = − 3 .
Calculate the sum of the factors: 16 + ( − 3 ) = 13 . The final answer is 13 ​ .

Explanation

Understanding the Problem We are looking for two factors of -48 that have a difference of 19. Let's call these factors x and y . We know that x × y = − 48 and x − y = 19 . Also, the factor with the greater absolute value is positive, which means 0"> x > 0 and |y|"> ∣ x ∣ > ∣ y ∣ .

Setting up the Equations From the equation x − y = 19 , we can express x in terms of y as x = y + 19 . Now, substitute this expression into the equation x × y = − 48 to get ( y + 19 ) × y = − 48 .

Forming the Quadratic Equation Expanding the equation, we have y 2 + 19 y = − 48 . Rearranging the terms, we get a quadratic equation: y 2 + 19 y + 48 = 0 .

Solving the Quadratic Equation Now, we need to solve this quadratic equation for y . We can factor the quadratic equation as ( y + 16 ) ( y + 3 ) = 0 . This gives us two possible values for y : y = − 16 or y = − 3 .

Finding the Factors If y = − 16 , then x = y + 19 = − 16 + 19 = 3 . If y = − 3 , then x = y + 19 = − 3 + 19 = 16 . Since we are given that the factor with the greater absolute value is positive, we must have 0"> x > 0 . In the first case, x = 3 and y = − 16 . In the second case, x = 16 and y = − 3 . Since |-3|"> ∣16∣ > ∣ − 3∣ , the correct pair of factors is x = 16 and y = − 3 .

Calculating the Sum of the Factors Finally, we need to find the sum of the factors x + y . Since x = 16 and y = − 3 , the sum is 16 + ( − 3 ) = 13 .

Final Answer Therefore, the sum of the factors is 13.


Examples
Understanding factors and their sums is crucial in many areas, such as cryptography, where prime factorization plays a key role in securing data. For instance, the security of RSA encryption relies on the difficulty of factoring large numbers into their prime factors. By understanding how factors interact, students can appreciate the underlying principles that protect our digital communications.

Answered by GinnyAnswer | 2025-07-03

The two factors of -48 that have a difference of 19 are 16 and -3. The sum of these factors is 13. Therefore, the answer is 13.
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Answered by Anonymous | 2025-07-04