what multiplies to -30 and adds to -7

Natasha willow

2 min read

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16-01-2025

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This article will guide you step-by-step through solving the classic math problem: finding two numbers that multiply to -30 and add up to -7. We'll explore different methods, from trial and error to a more systematic approach using algebra. Understanding this concept is crucial for factoring quadratic equations and solving various algebraic problems.

Understanding the Problem

The problem asks us to find two numbers, let's call them 'x' and 'y', that satisfy two conditions:

• x * y = -30: The product of the two numbers is -30.
• x + y = -7: The sum of the two numbers is -7.

Let's explore the solutions.

Method 1: Trial and Error

This method involves listing the pairs of factors of -30 and checking their sums. Since the product is negative, one number must be positive and the other negative.

Let's list the factor pairs of -30:

• 1 and -30 (sum: -29)
• 2 and -15 (sum: -13)
• 3 and -10 (sum: -7) This is our solution!
• 5 and -6 (sum: -1)
• 6 and -5 (sum: 1)
• 10 and -3 (sum: 7)
• 15 and -2 (sum: 13)
• 30 and -1 (sum: 29)

Therefore, the two numbers are 3 and -10.

Method 2: Using Algebra

We can solve this problem algebraically by setting up a system of two equations:

• Equation 1: x * y = -30
• Equation 2: x + y = -7

We can solve for one variable in terms of the other. Let's solve Equation 2 for x:

x = -7 - y

Now, substitute this expression for x into Equation 1:

(-7 - y) * y = -30

Expand and rearrange the equation:

-7y - y² = -30 y² + 7y - 30 = 0

This is a quadratic equation. We can solve it by factoring:

(y + 10)(y - 3) = 0

This gives us two possible solutions for y:

• y = -10
• y = 3

Now, substitute these values back into the equation x = -7 - y to find the corresponding values of x:

• If y = -10, then x = -7 - (-10) = 3
• If y = 3, then x = -7 - 3 = -10

Again, we find that the two numbers are 3 and -10.

Understanding the Signs

The fact that the product is negative and the sum is negative tells us that the larger number in absolute value must be negative. This helps to narrow down the possibilities when using the trial and error method.

Conclusion

Both the trial and error method and the algebraic method confirm that the two numbers that multiply to -30 and add up to -7 are 3 and -10. Understanding these methods is fundamental to solving similar problems in algebra and beyond. Remember to always check your answers by verifying that they satisfy both conditions of the problem.