The Product Of Two Positive Numbers Is 1,800. One Number Is Twice The Other Number. The Equation For The Two Numbers, With $x$ Representing The Lesser Number, Is $x(2x)=1,800$.What Is The Value Of The Lesser Number?A. 24 B. 30
Introduction
In mathematics, solving equations is a fundamental concept that helps us understand various relationships between numbers. In this article, we will delve into a problem involving two positive numbers whose product is 1,800. One number is twice the other, and we need to find the value of the lesser number. This problem can be represented by the equation $x(2x)=1,800$, where $x$ represents the lesser number. Our goal is to solve for the value of $x$.
Understanding the Equation
The given equation is $x(2x)=1,800$. To simplify this equation, we can expand the left-hand side by multiplying $x$ by $2x$, which gives us $2x^2=1,800$. This equation represents a quadratic relationship between the variable $x$ and the constant 1,800.
Solving the Quadratic Equation
To solve the quadratic equation $2x^2=1,800$, we need to isolate the variable $x$. We can start by dividing both sides of the equation by 2, which gives us $x^2=900$. This equation represents a perfect square, and we can take the square root of both sides to find the value of $x$.
Finding the Value of x
Taking the square root of both sides of the equation $x^2=900$, we get $x=\pm\sqrt{900}$. Since we are dealing with positive numbers, we can ignore the negative solution and focus on the positive square root. Evaluating the square root of 900, we get $x=\sqrt{900}=30$. Therefore, the value of the lesser number is 30.
Conclusion
In this article, we explored a mathematical problem involving two positive numbers whose product is 1,800. One number is twice the other, and we needed to find the value of the lesser number. By representing the problem as a quadratic equation and solving for the variable $x$, we found that the value of the lesser number is 30.
Additional Insights
- The given equation $x(2x)=1,800$ can be rewritten as $2x^2=1,800$, which represents a quadratic relationship between the variable $x$ and the constant 1,800.
- To solve the quadratic equation $2x^2=1,800$, we need to isolate the variable $x$ by dividing both sides of the equation by 2.
- Taking the square root of both sides of the equation $x^2=900$, we get $x=\pm\sqrt{900}$. Since we are dealing with positive numbers, we can ignore the negative solution and focus on the positive square root.
- Evaluating the square root of 900, we get $x=\sqrt{900}=30$. Therefore, the value of the lesser number is 30.
Frequently Asked Questions
- What is the product of the two positive numbers?
- The product of the two positive numbers is 1,800.
- What is the relationship between the two numbers?
- One number is twice the other number.
- What is the value of the lesser number?
- The value of the lesser number is 30.
Final Thoughts
In conclusion, solving the equation $x(2x)=1,800$ requires us to understand the relationship between the variable $x$ and the constant 1,800. By representing the problem as a quadratic equation and solving for the variable $x$, we found that the value of the lesser number is 30. This problem demonstrates the importance of algebraic techniques in solving mathematical equations and understanding various relationships between numbers.
Introduction
In our previous article, we explored a mathematical problem involving two positive numbers whose product is 1,800. One number is twice the other, and we needed to find the value of the lesser number. In this article, we will provide a Q&A section to address common questions and provide additional insights into the problem.
Q&A Section
Q: What is the product of the two positive numbers?
A: The product of the two positive numbers is 1,800.
Q: What is the relationship between the two numbers?
A: One number is twice the other number.
Q: How do we represent the problem as an equation?
A: We can represent the problem as the equation $x(2x)=1,800$, where $x$ represents the lesser number.
Q: How do we simplify the equation?
A: We can simplify the equation by expanding the left-hand side and getting $2x^2=1,800$.
Q: How do we solve the quadratic equation?
A: We can solve the quadratic equation by dividing both sides of the equation by 2 and getting $x^2=900$. Then, we can take the square root of both sides to find the value of $x$.
Q: What is the value of the lesser number?
A: The value of the lesser number is 30.
Q: Why do we ignore the negative solution?
A: We ignore the negative solution because we are dealing with positive numbers.
Q: How do we evaluate the square root of 900?
A: We can evaluate the square root of 900 by getting $x=\sqrt{900}=30$.
Q: What is the final answer?
A: The final answer is 30.
Additional Insights
- The given equation $x(2x)=1,800$ can be rewritten as $2x^2=1,800$, which represents a quadratic relationship between the variable $x$ and the constant 1,800.
- To solve the quadratic equation $2x^2=1,800$, we need to isolate the variable $x$ by dividing both sides of the equation by 2.
- Taking the square root of both sides of the equation $x^2=900$, we get $x=\pm\sqrt{900}$. Since we are dealing with positive numbers, we can ignore the negative solution and focus on the positive square root.
- Evaluating the square root of 900, we get $x=\sqrt{900}=30$. Therefore, the value of the lesser number is 30.
Frequently Asked Questions
- What is the product of the two positive numbers?
- The product of the two positive numbers is 1,800.
- What is the relationship between the two numbers?
- One number is twice the other number.
- What is the value of the lesser number?
- The value of the lesser number is 30.
Final Thoughts
In conclusion, solving the equation $x(2x)=1,800$ requires us to understand the relationship between the variable $x$ and the constant 1,800. By representing the problem as a quadratic equation and solving for the variable $x$, we found that the value of the lesser number is 30. This problem demonstrates the importance of algebraic techniques in solving mathematical equations and understanding various relationships between numbers.
Related Topics
- Quadratic equations
- Algebraic techniques
- Mathematical relationships
- Problem-solving strategies
Resources
Conclusion
In this Q&A article, we provided additional insights into the problem of finding the value of the lesser number. We addressed common questions and provided step-by-step solutions to the quadratic equation. We hope that this article has been helpful in understanding the problem and its solution.