what two numbers multiply to be -56 and add up to be -1?
The Correct Answer and Explanation is:
Step 1: Understand the problem
We are given two conditions:
- The product of the two numbers is –56.
- The sum of the two numbers is –1.
This tells us we are looking for two integers that, when multiplied together, give –56, and when added together, give –1.
Step 2: List factor pairs of –56
Because the product is negative, one number must be positive and the other negative. Here are the factor pairs of 56:
- 1 × 56
- 2 × 28
- 4 × 14
- 7 × 8
Now apply negative signs to one number in each pair to get possible combinations for –56:
- –1 and 56 (sum = 55)
- 1 and –56 (sum = –55)
- –2 and 28 (sum = 26)
- 2 and –28 (sum = –26)
- –4 and 14 (sum = 10)
- 4 and –14 (sum = –10)
- –7 and 8 (sum = 1)
- 7 and –8 (sum = –1) ← This is the correct pair!
Final Answer:
The two numbers are 7 and –8.
Explanation (Textbook Style – 300 Words)
To solve this problem, we need to find two numbers that multiply to –56 and add up to –1. This type of problem often appears in algebra when factoring trinomials, solving quadratic equations, or working with number puzzles.
The product being negative tells us that one number must be positive and the other negative. That’s because only unlike signs produce a negative result when multiplying. The sum of the numbers is –1, which means the positive number must be slightly smaller in absolute value than the negative number.
Start by listing factor pairs of 56, since 56 is the absolute value of the product. These include (1, 56), (2, 28), (4, 14), and (7, 8). Next, test each factor pair with opposite signs to check which combination gives a sum of –1.
Trying (7, –8), the multiplication is:
7 × (–8) = –56 ✅
and the addition is:
7 + (–8) = –1 ✅
This confirms that 7 and –8 are the correct pair. No other factor combination will satisfy both conditions.
This process shows the importance of understanding both signs and operations when solving number problems. Mastering how to manipulate integers and work with factor pairs is a crucial foundation for higher-level algebra, including solving equations and factoring polynomials.
