What 2 Numbers Multiply To -108 And Add To -15

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Introduction

Mathematics is a fascinating subject that involves the study of numbers, quantities, and shapes. It is a fundamental subject that is used in various aspects of life, including science, technology, engineering, and mathematics (STEM). In mathematics, there are various types of problems that require different approaches to solve. In this article, we will focus on a specific problem that involves finding two numbers that multiply to -108 and add to -15.

The Problem

The problem states that we need to find two numbers that multiply to -108 and add to -15. This is a classic problem in algebra, and it requires us to use our knowledge of equations and inequalities to solve. The problem can be represented mathematically as:

x * y = -108 x + y = -15

Understanding the Problem

To solve this problem, we need to understand the relationship between the two equations. We can start by analyzing the first equation, which states that the product of x and y is equal to -108. This means that either x or y (or both) must be negative, since the product of two positive numbers cannot be negative.

Using Algebraic Techniques

One way to solve this problem is to use algebraic techniques. We can start by isolating one of the variables in the second equation. Let's isolate x by subtracting y from both sides of the equation:

x = -15 - y

Substituting into the First Equation

Now that we have isolated x, we can substitute this expression into the first equation:

(-15 - y) * y = -108

Expanding the Equation

To simplify the equation, we can expand the left-hand side by multiplying the two terms:

-15y - y^2 = -108

Rearranging the Equation

To make it easier to solve, we can rearrange the equation by moving all the terms to one side:

y^2 + 15y - 108 = 0

Solving the Quadratic Equation

This is a quadratic equation, and we can solve it using the quadratic formula or by factoring. Let's try to factor the equation:

(y + 18)(y - 6) = 0

Finding the Solutions

To find the solutions, we can set each factor equal to zero and solve for y:

y + 18 = 0 --> y = -18 y - 6 = 0 --> y = 6

Finding the Corresponding Values of x

Now that we have found the values of y, we can find the corresponding values of x by substituting these values into the expression we derived earlier:

x = -15 - y

For y = -18, we have:

x = -15 - (-18) = -15 + 18 = 3

For y = 6, we have:

x = -15 - 6 = -21

Conclusion

In this article, we have solved the problem of finding two numbers that multiply to -108 and add to -15. We used algebraic techniques to isolate one of the variables and then substituted this expression into the first equation. We then solved the resulting quadratic equation by factoring and found the corresponding values of x. The two numbers that satisfy the given conditions are x = 3 and y = -18, or x = -21 and y = 6.

Additional Tips and Tricks

  • When solving quadratic equations, it's often helpful to try to factor the equation before using the quadratic formula.
  • When substituting expressions into equations, make sure to simplify the resulting equation before solving.
  • When solving systems of equations, it's often helpful to use substitution or elimination methods to isolate one of the variables.

Frequently Asked Questions

  • Q: What is the product of the two numbers that multiply to -108 and add to -15? A: The product of the two numbers is -108.
  • Q: What is the sum of the two numbers that multiply to -108 and add to -15? A: The sum of the two numbers is -15.
  • Q: How do I solve a quadratic equation? A: You can solve a quadratic equation by factoring, using the quadratic formula, or by completing the square.

Final Thoughts

Mathematics is a fascinating subject that involves the study of numbers, quantities, and shapes. In this article, we have solved the problem of finding two numbers that multiply to -108 and add to -15. We used algebraic techniques to isolate one of the variables and then substituted this expression into the first equation. We then solved the resulting quadratic equation by factoring and found the corresponding values of x. The two numbers that satisfy the given conditions are x = 3 and y = -18, or x = -21 and y = 6.

Introduction

In our previous article, we solved the problem of finding two numbers that multiply to -108 and add to -15. We used algebraic techniques to isolate one of the variables and then substituted this expression into the first equation. We then solved the resulting quadratic equation by factoring and found the corresponding values of x. In this article, we will answer some of the most frequently asked questions related to this problem.

Q&A

Q: What is the product of the two numbers that multiply to -108 and add to -15?

A: The product of the two numbers is -108.

Q: What is the sum of the two numbers that multiply to -108 and add to -15?

A: The sum of the two numbers is -15.

Q: How do I solve a quadratic equation?

A: You can solve a quadratic equation by factoring, using the quadratic formula, or by completing the square.

Q: What is the difference between the two numbers that multiply to -108 and add to -15?

A: The difference between the two numbers is 21 (since 3 - (-18) = 21 or -21 - 6 = 21).

Q: Can I use a calculator to solve this problem?

A: Yes, you can use a calculator to solve this problem. However, it's often more helpful to understand the underlying math and use algebraic techniques to solve the problem.

Q: How do I know which numbers to use when solving a quadratic equation?

A: When solving a quadratic equation, you can use the quadratic formula or try to factor the equation. If you're unable to factor the equation, you can use the quadratic formula to find the solutions.

Q: What is the relationship between the two numbers that multiply to -108 and add to -15?

A: The two numbers are related by the fact that their product is -108 and their sum is -15.

Q: Can I use this method to solve other problems involving quadratic equations?

A: Yes, you can use this method to solve other problems involving quadratic equations. The key is to understand the underlying math and use algebraic techniques to solve the problem.

Q: How do I check my answers when solving a quadratic equation?

A: When solving a quadratic equation, it's always a good idea to check your answers by plugging them back into the original equation. This will help you ensure that your solutions are correct.

Q: What is the significance of the quadratic formula in solving quadratic equations?

A: The quadratic formula is a powerful tool for solving quadratic equations. It allows you to find the solutions to a quadratic equation in a straightforward and efficient manner.

Q: Can I use the quadratic formula to solve other types of equations?

A: No, the quadratic formula is specifically designed to solve quadratic equations. However, you can use other methods, such as factoring or completing the square, to solve other types of equations.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the problem of finding two numbers that multiply to -108 and add to -15. We have provided explanations and examples to help you understand the underlying math and use algebraic techniques to solve the problem. Whether you're a student or a teacher, we hope this article has been helpful in providing you with a deeper understanding of quadratic equations and how to solve them.

Additional Resources

  • For more information on quadratic equations, check out our article on "Solving Quadratic Equations: A Step-by-Step Guide".
  • For more practice problems on quadratic equations, check out our article on "Quadratic Equation Practice Problems".
  • For more information on algebraic techniques, check out our article on "Algebraic Techniques for Solving Equations".

Final Thoughts

Mathematics is a fascinating subject that involves the study of numbers, quantities, and shapes. In this article, we have provided explanations and examples to help you understand the underlying math and use algebraic techniques to solve the problem of finding two numbers that multiply to -108 and add to -15. Whether you're a student or a teacher, we hope this article has been helpful in providing you with a deeper understanding of quadratic equations and how to solve them.