what multiplies to -30 but adds to 1

Natashya

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27-12-2024

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Finding two numbers that satisfy these conditions involves a bit of number sense and perhaps some trial and error. Let's explore how to solve this common math problem.

Understanding the Problem

The problem presents two key pieces of information:

1. The product: The two numbers multiply to -30. This means one number is positive, and one is negative.
2. The sum: The two numbers add up to 1. This indicates the positive number is larger in magnitude than the negative number.

Method 1: Listing Factors

One straightforward approach is to list the factor pairs of -30:

• -1 and 30
• -2 and 15
• -3 and 10
• -5 and 6
• -6 and 5
• -10 and 3
• -15 and 2
• -30 and 1

Now, let's check which pair adds up to 1:

Only -5 and 6 satisfy both conditions. -5 + 6 = 1, and -5 * 6 = -30.

Method 2: Algebraic Approach

We can also solve this using algebra. Let's represent the two numbers as 'x' and 'y'. We can set up two equations based on the given information:

• Equation 1 (Product): x * y = -30
• Equation 2 (Sum): x + y = 1

We can solve for one variable in Equation 2 and substitute it into Equation 1. Let's solve for 'y':

y = 1 - x

Now substitute this into Equation 1:

x * (1 - x) = -30

Expanding and rearranging, we get a quadratic equation:

x² - x - 30 = 0

This quadratic equation can be factored as:

(x - 6)(x + 5) = 0

This gives us two possible solutions for x:

• x = 6
• x = -5

If x = 6, then y = 1 - 6 = -5. If x = -5, then y = 1 - (-5) = 6.

Both solutions lead to the same pair of numbers: 6 and -5.

Conclusion

Therefore, the two numbers that multiply to -30 and add to 1 are 6 and -5. This problem highlights the importance of understanding the relationship between factors, products, and sums in number theory. Both the listing factors and the algebraic methods provide effective ways to arrive at the solution.