The Product Of Two Consecutive Negative Integers Is 600. What Is The Value Of The Lesser Integer?A. \[$-60\$\]B. \[$-30\$\]C. \[$-25\$\]D. \[$-15\$\]

Introduction

In mathematics, problems involving consecutive integers often require creative thinking and a deep understanding of number properties. This article will delve into a specific problem where the product of two consecutive negative integers is given as 600, and we need to find the value of the lesser integer.

Understanding the Problem

Let's denote the lesser integer as $n$. Since the integers are consecutive, the greater integer can be represented as $n + 1$. The product of these two integers is given as 600, which can be expressed as:

$$n(n + 1) = 600$$

Our goal is to find the value of the lesser integer, $n$.

Solving the Equation

To solve this equation, we can start by expanding the product:

$$n^2 + n - 600 = 0$$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = 1$, and $c = -600$. We can use the quadratic formula to find the solutions:

$$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting the values of $a$, $b$, and $c$, we get:

$$n = \frac{-1 \pm \sqrt{1 + 2400}}{2}$$

Simplifying the expression under the square root, we get:

$$n = \frac{-1 \pm \sqrt{2401}}{2}$$

Since $\sqrt{2401} = 49$, we can further simplify the expression:

$$n = \frac{-1 \pm 49}{2}$$

This gives us two possible solutions for $n$:

$$n = \frac{-1 + 49}{2} = 24$$

$$n = \frac{-1 - 49}{2} = -25$$

However, we are looking for the lesser integer, which is a negative number. Therefore, the correct solution is $n = -25$.

Conclusion

In this article, we have solved a mathematical puzzle involving the product of two consecutive negative integers. By using the quadratic formula and simplifying the expression, we have found the value of the lesser integer to be $-25$. This problem requires a deep understanding of number properties and the ability to apply mathematical concepts to solve real-world problems.

Answer

The value of the lesser integer is $\boxed{-25}$.

Additional Tips and Tricks

When solving problems involving consecutive integers, it's essential to remember that the integers are always separated by a difference of 1. This can help you identify the correct solution and avoid common pitfalls.

In this problem, we used the quadratic formula to find the solutions. However, there are other methods to solve quadratic equations, such as factoring or using the quadratic formula with a calculator. The choice of method depends on the specific problem and the level of difficulty.

Real-World Applications

This problem may seem abstract, but it has real-world applications in various fields, such as:

• Finance: When calculating interest rates or investment returns, consecutive integers can be used to model the growth or decline of an investment.
• Science: In physics, consecutive integers can be used to model the motion of objects or the behavior of particles.
• Engineering: In computer science, consecutive integers can be used to model the behavior of algorithms or the performance of systems.

By understanding the properties of consecutive integers, we can develop a deeper appreciation for the mathematical concepts that underlie many real-world problems.

Final Thoughts

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Introduction

In our previous article, we solved a mathematical puzzle involving the product of two consecutive negative integers. We found that the value of the lesser integer is $-25$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the problem.

Q&A

Q: What is the product of two consecutive negative integers?

A: The product of two consecutive negative integers is given as 600.

Q: How do we represent the lesser and greater integers?

A: Let's denote the lesser integer as $n$. Since the integers are consecutive, the greater integer can be represented as $n + 1$.

Q: What is the equation that represents the product of the two integers?

A: The equation that represents the product of the two integers is:

$$n(n + 1) = 600$$

Q: How do we solve the equation?

A: We can start by expanding the product:

$$n^2 + n - 600 = 0$$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = 1$, and $c = -600$. We can use the quadratic formula to find the solutions:

$$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: What are the possible solutions for $n$?

A: Substituting the values of $a$, $b$, and $c$, we get:

$$n = \frac{-1 \pm \sqrt{1 + 2400}}{2}$$

Simplifying the expression under the square root, we get:

$$n = \frac{-1 \pm \sqrt{2401}}{2}$$

Since $\sqrt{2401} = 49$, we can further simplify the expression:

$$n = \frac{-1 \pm 49}{2}$$

This gives us two possible solutions for $n$:

$$n = \frac{-1 + 49}{2} = 24$$

$$n = \frac{-1 - 49}{2} = -25$$

Q: Which solution is the correct value of the lesser integer?

A: Since we are looking for the lesser integer, which is a negative number, the correct solution is $n = -25$.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as finance, science, and engineering. For example, in finance, consecutive integers can be used to model the growth or decline of an investment. In science, consecutive integers can be used to model the motion of objects or the behavior of particles.

Q: What are some tips and tricks for solving problems involving consecutive integers?

A: When solving problems involving consecutive integers, it's essential to remember that the integers are always separated by a difference of 1. This can help you identify the correct solution and avoid common pitfalls.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional insights into the problem of the product of two consecutive negative integers. We have found that the value of the lesser integer is $-25$. This problem requires a combination of mathematical knowledge and critical thinking skills. By understanding the properties of consecutive integers, we can develop a deeper appreciation for the mathematical concepts that underlie many real-world problems.

Additional Resources

For more information on quadratic equations and consecutive integers, please refer to the following resources:

• Mathematics textbooks: Many mathematics textbooks cover quadratic equations and consecutive integers in detail.
• Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and exercises on quadratic equations and consecutive integers.
• Mathematical software: Software such as Mathematica, Maple, and MATLAB can be used to solve quadratic equations and explore the properties of consecutive integers.

Final Thoughts

In conclusion, this problem requires a combination of mathematical knowledge and critical thinking skills. By breaking down the problem into smaller steps and using the quadratic formula, we have found the value of the lesser integer to be $-25$. This problem serves as a reminder of the importance of mathematical thinking and problem-solving skills in real-world applications.