What two numbers multiply to get -144 and add to get 0?
The Correct Answer and Explanation is :
To find two numbers that multiply to (-144) and add to (0), we can denote the numbers as (x) and (y). From the problem, we have the following equations:
- (x + y = 0)
- (xy = -144)
From the first equation, we can express (y) in terms of (x):
[ y = -x ]
Next, we substitute this expression for (y) into the second equation:
[ x(-x) = -144 ]
This simplifies to:
[ -x^2 = -144 ]
By multiplying both sides by (-1), we have:
[ x^2 = 144 ]
Taking the square root of both sides gives us two possible values for (x):
[ x = 12 \quad \text{or} \quad x = -12 ]
Now, substituting these values back into the expression for (y):
- If (x = 12), then (y = -12).
- If (x = -12), then (y = 12).
Thus, the two numbers we are looking for are (12) and (-12).
Explanation
To understand why these numbers meet the criteria, we analyze both conditions:
- Multiplication to -144: When you multiply (12) and (-12), the result is:
[
12 \times (-12) = -144
]
This confirms that they indeed multiply to (-144). - Addition to 0: Adding these two numbers results in:
[
12 + (-12) = 0
]
This shows that they add up to (0).
Conclusion
The solution highlights an important aspect of algebra: relationships between numbers can often be defined in two ways—through their product and their sum. In this case, (12) and (-12) are not only a perfect example of opposite numbers but also illustrate how two numbers can satisfy conditions that at first glance might seem contradictory, like multiplying to a negative number while summing to zero.