what adds to -2 and muliplies to -35

mestopa

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21-01-2025

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Finding two numbers that meet specific addition and multiplication criteria is a common algebra problem. This article will walk you through solving the question: "What two numbers add up to -2 and multiply to -35?" We'll explore different approaches, from guess-and-check to a more systematic algebraic method.

Understanding the Problem

The problem presents two key pieces of information:

• Sum: The two numbers add up to -2.
• Product: The two numbers multiply to -35.

Our goal is to find the pair of numbers that satisfy both conditions simultaneously.

Method 1: Trial and Error (Guess and Check)

One approach is to try different pairs of numbers. Since the product is negative (-35), we know one number must be positive and the other negative. Let's start by considering the factors of -35:

• -1 and 35
• -5 and 7
• -7 and 5
• -35 and 1

Now let's check which pair adds up to -2:

• -1 + 35 = 34 (Incorrect)
• -5 + 7 = 2 (Incorrect)
• -7 + 5 = -2 (Correct!)
• -35 + 1 = -34 (Incorrect)

Therefore, the two numbers are -7 and 5.

Method 2: Using Algebra

A more systematic approach uses algebra. Let's represent the two numbers as 'x' and 'y'. We can then set up a system of two equations based on the given information:

• Equation 1 (Sum): x + y = -2
• Equation 2 (Product): x * y = -35

We can solve this system of equations using substitution or elimination. Let's use substitution:

1. Solve Equation 1 for one variable: Let's solve for 'x': x = -2 - y

2. Substitute: Substitute this expression for 'x' into Equation 2: (-2 - y) * y = -35

3. Expand and simplify: -2y - y² = -35 => y² + 2y - 35 = 0

4. Factor the quadratic equation: (y + 7)(y - 5) = 0

5. Solve for 'y': This gives us two possible solutions for 'y': y = -7 or y = 5

6. Solve for 'x': Substitute each value of 'y' back into the equation x = -2 - y:

   • If y = -7, then x = -2 - (-7) = 5
   • If y = 5, then x = -2 - 5 = -7

Again, we find that the two numbers are -7 and 5.

Conclusion

Both the trial-and-error and algebraic methods confirm that the two numbers that add up to -2 and multiply to -35 are -7 and 5. Choosing the best method depends on your comfort level with algebra and the complexity of the problem. For simple problems like this, guess-and-check might be quicker. However, for more complex scenarios, the algebraic approach provides a more reliable and systematic way to find the solution.