Solving for Multiplication and Addition of Two Numbers With Specific Constraints

Mathematics often involves solving equations to find the values of unknown variables. This article will guide you through a specific problem where you need to find two numbers that, when multiplied together, give a negative product, and when added together, give a positive sum. This type of problem can be encountered in various fields, including cryptography, physics, and economics.

Problem Statement

Question: What two numbers, when multiplied together, give a negative 24, and when added together, give a positive 5?

Step-by-Step Solution

Let's denote the two numbers as x and y. We need to satisfy the following two conditions:

xy -24 x y 5

Factor Pairs of -24

The factor pairs of -24 are:

-1 times; 24 -2 times; 12 -3 times; 8 -4 times; 6

We need to find a pair where one number is positive and the other is negative, and their absolute values differ such that when added together, they give 5. Let's analyze each pair:

-1 24 23 (not 5) -2 12 10 (not 5) -3 8 5 (this works) -4 6 2 (not 5)

The pair that satisfies both conditions is -3 and 8. Therefore, x -3 and y 8.

Verification

Let's verify the solution by plugging the values into the equations:

(-3) times; 8 -24 (satisfies xy -24) -3 8 5 (satisfies x y 5)

The solution is confirmed to be correct.

Algebraic Method

We can also solve this problem using an algebraic approach. Let's denote the two numbers as x and y. We have the following equations:

xy -24 x y 5

Solving for y

From the second equation, we can express y in terms of x:

y 5 - x

Substituting and Solving

Substitute this expression for y into the first equation:

x(5 - x) -24

Expanding and rearranging the equation:

5x - x2 -24

x2 - 5x - 24 0

Factorizing the quadratic equation:

(x - 8)(x 3) 0

Therefore, we have two solutions:

x 8 or x -3

If x 8, then y 5 - 8 -3.

If x -3, then y 5 - (-3) 8.

So, the two numbers are 8 and -3.

Conclusion

The two numbers that satisfy the conditions are 8 and -3. These numbers multiply to give -24 and add to give 5. This method can be applied to similar problems to find the correct values of the unknowns.