The Product Of Two Consecutive Positive Integers Is 812. What Is The Value Of The Lesser Integer?
Introduction
In mathematics, problems involving consecutive integers are often used to test a student's understanding of algebraic expressions and equations. In this article, we will explore a classic problem that involves finding the value of the lesser integer when the product of two consecutive positive integers is given as 812. We will use algebraic methods to solve this problem and provide a step-by-step solution.
Understanding the Problem
Let's assume that the two consecutive positive integers are x and x+1. We are given that their product is 812, which can be expressed as:
x(x+1) = 812
Our goal is to find the value of x, which represents the lesser integer.
Solving the Equation
To solve this equation, we can start by expanding the left-hand side:
x^2 + x - 812 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 1, and c = -812. We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
x = (-(1) ± √((1)^2 - 4(1)(-812))) / 2(1) x = (-1 ± √(1 + 3248)) / 2 x = (-1 ± √3249) / 2 x = (-1 ± 57) / 2
We have two possible solutions for x:
x = (-1 + 57) / 2 = 28 x = (-1 - 57) / 2 = -29
However, we are looking for a positive integer, so we discard the negative solution.
The Value of the Lesser Integer
Therefore, the value of the lesser integer is x = 28.
Conclusion
In this article, we used algebraic methods to solve a classic problem involving the product of two consecutive positive integers. We started by expressing the problem as a quadratic equation and then used the quadratic formula to find the value of the lesser integer. The final answer is x = 28.
Additional Tips and Tricks
- When solving quadratic equations, it's essential to check the solutions to ensure they are valid in the context of the problem.
- In this case, we discarded the negative solution because we were looking for a positive integer.
- The quadratic formula can be used to solve quadratic equations in the form ax^2 + bx + c = 0, where a, b, and c are constants.
Real-World Applications
This problem 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 of an investment over time.
- Science: In physics, consecutive integers can be used to model the motion of objects, such as the position and velocity of a particle.
- Engineering: In engineering, consecutive integers can be used to model the behavior of complex systems, such as the vibration of a mechanical system.
Final Thoughts
Introduction
In our previous article, we explored a classic problem involving the product of two consecutive positive integers. We used algebraic methods to solve the problem and found the value of the lesser integer. In this article, we will provide a Q&A section to address common questions and concerns related to this problem.
Q&A Section
Q: What is the product of two consecutive positive integers? A: The product of two consecutive positive integers is a mathematical expression that represents the result of multiplying two consecutive integers together. For example, if we have two consecutive integers 3 and 4, their product is 3 × 4 = 12.
Q: How do I solve the equation x(x+1) = 812? A: To solve the equation x(x+1) = 812, we can start by expanding the left-hand side to get x^2 + x - 812 = 0. We can then use the quadratic formula to find the value of x.
Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that is used to solve quadratic equations in the form ax^2 + bx + c = 0. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: Why do I get two possible solutions for x? A: When using the quadratic formula, we get two possible solutions for x because the formula involves a ± symbol. This means that we have two possible values for x, which are x = (-b + √(b^2 - 4ac)) / 2a and x = (-b - √(b^2 - 4ac)) / 2a.
Q: How do I determine which solution is valid? A: To determine which solution is valid, we need to check the solutions to ensure they are valid in the context of the problem. In this case, we discarded the negative solution because we were looking for a positive integer.
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 of an investment over time. In science, consecutive integers can be used to model the motion of objects, such as the position and velocity of a particle.
Q: Can I use this problem to model complex systems? A: Yes, this problem can be used to model complex systems, such as the vibration of a mechanical system. By using consecutive integers to model the behavior of the system, we can gain a better understanding of how the system behaves over time.
Q: Are there any other ways to solve this problem? A: Yes, there are other ways to solve this problem, such as using factoring or the quadratic formula. However, the quadratic formula is often the most efficient method for solving quadratic equations.
Conclusion
In this article, we provided a Q&A section to address common questions and concerns related to the product of two consecutive positive integers. We covered topics such as the quadratic formula, real-world applications, and alternative methods for solving the problem. By understanding these concepts, we can gain a better appreciation for the mathematical beauty of this problem.
Additional Tips and Tricks
- When solving quadratic equations, it's essential to check the solutions to ensure they are valid in the context of the problem.
- In this case, we discarded the negative solution because we were looking for a positive integer.
- The quadratic formula can be used to solve quadratic equations in the form ax^2 + bx + c = 0, where a, b, and c are constants.
Real-World Applications
This problem 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 of an investment over time.
- Science: In physics, consecutive integers can be used to model the motion of objects, such as the position and velocity of a particle.
- Engineering: In engineering, consecutive integers can be used to model the behavior of complex systems, such as the vibration of a mechanical system.
Final Thoughts
In conclusion, the product of two consecutive positive integers is a classic problem that can be solved using algebraic methods. By using the quadratic formula, we can find the value of the lesser integer, which is x = 28. This problem has real-world applications in various fields and can be used to model complex systems and behaviors.