What multiplies to 150 and adds to -31 – Exploring the fascinating world of mathematics, we embark on a journey to unravel the secrets of numbers that multiply to 150 and add to -31. This intriguing problem invites us to delve into the depths of multiplication and addition, revealing the intricate relationships between numbers.
Our quest begins by understanding the fundamental concepts of multiplication and addition. We will then embark on a systematic approach to solving the system of equations that governs this problem, guiding you through each step with clarity and precision.
Identifying Factors: What Multiplies To 150 And Adds To -31
Multiplication and addition are two fundamental mathematical operations. Multiplication involves combining equal groups to find the total, while addition involves combining two or more numbers to find their sum. To find two numbers that multiply to 150 and add to -31, we need to understand the relationship between these operations.
When two numbers are multiplied, their product is the result of adding one number to itself as many times as the other number. For example, 6 multiplied by 5 is the same as adding 6 to itself 5 times, which equals 30. When two numbers are added, their sum is the result of combining their values.
For example, 6 added to 5 is 11.
To find two numbers that multiply to 150 and add to -31, we need to consider the factors of 150 and the possible combinations of those factors that add up to -31. Factors are the numbers that, when multiplied together, produce a given number.
The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, and 150.
Solving Equations
We can set up a system of equations to find the two numbers that multiply to 150 and add to – 31. Let x and y represent the two numbers. Then, we have the following equations:
- x – y = 150
- x + y = -31
We can solve this system of equations by substitution. First, we solve the second equation for x:
- x = -31 – y
Then, we substitute this expression for x into the first equation:
- (-31 – y) – y = 150
We can now solve this equation for y by expanding the left side and simplifying:
- -31y – y^2 = 150
- y^2 + 31y + 150 = 0
- (y + 15)(y + 10) = 0
Therefore, y = -15 or y = -10. Substituting these values back into the expression for x, we get x = -16 or x = -21.
Examples and Illustrations
Here is an example of a pair of numbers that multiply to 150 and add to -31:
- x = -16
- y = -15
We can verify that these numbers satisfy both equations:
- x – y = (-16) – (-15) = 240
- x + y = (-16) + (-15) = -31
Here is a table of multiple examples of pairs of numbers that multiply to 150 and add to -31:
| x | y |
|---|---|
| -16 | -15 |
| -21 | -10 |
Extensions and Applications, What multiplies to 150 and adds to -31
The concept of finding two numbers that multiply to a given value and add to a given sum has applications in various fields, such as physics and engineering. For example, in physics, it can be used to find the velocity and acceleration of an object given its displacement and time.
The problem can also be extended to finding three or more numbers that multiply to a given value and add to a given sum. This problem is more challenging and requires more advanced mathematical techniques.
FAQ Insights
What is the significance of finding numbers that multiply to 150 and add to-31?
Understanding how to find numbers that satisfy specific mathematical relationships is a fundamental skill in mathematics. It has applications in various fields, including physics, engineering, and computer science.
Can we find more than two numbers that multiply to 150 and add to-31?
Yes, there are infinitely many pairs of numbers that satisfy this condition. However, finding all of them is a complex mathematical problem.
