what adds to -2 and muliplies to -35

Natashya

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02-02-2025

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Finding two numbers that satisfy specific addition and multiplication conditions is a common algebra problem. This article will guide you through solving the question: "What two numbers add to -2 and multiply to -35?" We'll explore multiple approaches, making this concept clear and accessible.

Understanding the Problem

The problem presents two equations:

• Equation 1 (Addition): x + y = -2
• Equation 2 (Multiplication): x * y = -35

Our goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously.

Method 1: Trial and Error (for smaller numbers)

With relatively small numbers like -35, a trial-and-error approach can be effective. We need to consider pairs of factors of -35. Remember that one number must be positive and the other negative because their product is negative.

Let's list the factor pairs 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
• -7 + 5 = -2 (This is our solution!)

Therefore, the two numbers are -7 and 5.

Method 2: Using Algebra (for larger or more complex numbers)

For larger numbers or more complex problems, an algebraic approach is more efficient. We can use substitution or elimination. Let's use substitution:

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

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

3. Solve the quadratic equation: Expand and rearrange the equation: -2y - y² = -35 => y² + 2y - 35 = 0

4. Factor the quadratic: (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

Therefore, the two numbers are again -7 and 5.

Conclusion

Both methods lead to the same solution: the two numbers that add to -2 and multiply to -35 are -7 and 5. Choosing the best method depends on the complexity of the problem. Trial and error is quicker for simpler problems, while the algebraic approach is more robust and applicable to a wider range of scenarios. Understanding both methods provides you with a versatile toolkit for solving similar problems in algebra.