Solve For { X $}$ In The Equation:${ 1 = 3 + \sqrt{2x + 7} }$
Introduction to the Problem
When solving for x in an equation, it's essential to isolate the variable on one side of the equation. In this case, we have the equation 1 = 3 + โ(2x + 7), and our goal is to find the value of x. This equation involves a square root, which can be solved using algebraic techniques.
Understanding the Equation
The given equation is 1 = 3 + โ(2x + 7). To solve for x, we need to isolate the square root term. The first step is to subtract 3 from both sides of the equation, which gives us:
1 - 3 = โ(2x + 7)
Simplifying the Equation
Simplifying the left-hand side of the equation, we get:
-2 = โ(2x + 7)
Eliminating the Square Root
To eliminate the square root, we can square both sides of the equation. This will give us:
(-2)^2 = (โ(2x + 7))^2
Simplifying the Squared Terms
Simplifying the squared terms, we get:
4 = 2x + 7
Isolating the Variable
To isolate the variable x, we need to get rid of the constant term on the right-hand side of the equation. We can do this by subtracting 7 from both sides of the equation, which gives us:
4 - 7 = 2x
Simplifying the Equation
Simplifying the left-hand side of the equation, we get:
-3 = 2x
Solving for x
To solve for x, we need to divide both sides of the equation by 2, which gives us:
x = -3/2
Conclusion
In this article, we solved for x in the equation 1 = 3 + โ(2x + 7). We started by isolating the square root term, then squared both sides of the equation to eliminate the square root. Finally, we isolated the variable x and solved for its value. The solution to the equation is x = -3/2.
Step-by-Step Solution
Here's a step-by-step solution to the equation:
- Subtract 3 from both sides of the equation: 1 - 3 = โ(2x + 7)
- Simplify the left-hand side of the equation: -2 = โ(2x + 7)
- Square both sides of the equation: (-2)^2 = (โ(2x + 7))^2
- Simplify the squared terms: 4 = 2x + 7
- Subtract 7 from both sides of the equation: 4 - 7 = 2x
- Simplify the left-hand side of the equation: -3 = 2x
- Divide both sides of the equation by 2: x = -3/2
Tips and Tricks
When solving for x in an equation involving a square root, it's essential to isolate the square root term first. Then, you can square both sides of the equation to eliminate the square root. Finally, isolate the variable x and solve for its value.
Real-World Applications
Solving for x in an equation involving a square root has many real-world applications. For example, in physics, you may need to solve for the position of an object at a given time, which involves solving for x in an equation involving a square root. In engineering, you may need to solve for the stress on a material, which also involves solving for x in an equation involving a square root.
Common Mistakes
When solving for x in an equation involving a square root, there are several common mistakes to avoid. One mistake is to forget to isolate the square root term first. Another mistake is to square both sides of the equation without checking if the equation is valid. Finally, a common mistake is to forget to check the solution for extraneous solutions.
Conclusion
In conclusion, solving for x in the equation 1 = 3 + โ(2x + 7) involves isolating the square root term, squaring both sides of the equation, and isolating the variable x. By following these steps and avoiding common mistakes, you can solve for x in any equation involving a square root.
Q: What is the first step in solving for x in the equation 1 = 3 + โ(2x + 7)?
A: The first step in solving for x in the equation 1 = 3 + โ(2x + 7) is to isolate the square root term. This can be done by subtracting 3 from both sides of the equation.
Q: Why do we need to square both sides of the equation to eliminate the square root?
A: We need to square both sides of the equation to eliminate the square root because the square root is a non-linear operation. By squaring both sides of the equation, we can get rid of the square root and solve for x.
Q: What is the difference between an extraneous solution and a valid solution?
A: An extraneous solution is a solution that is not valid for the original equation. A valid solution, on the other hand, is a solution that satisfies the original equation. When solving for x in an equation involving a square root, it's essential to check for extraneous solutions.
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it's true. If the solution satisfies the original equation, then it's a valid solution. If it doesn't satisfy the original equation, then it's an extraneous solution.
Q: What are some common mistakes to avoid when solving for x in an equation involving a square root?
A: Some common mistakes to avoid when solving for x in an equation involving a square root include:
- Forgetting to isolate the square root term first
- Squaring both sides of the equation without checking if the equation is valid
- Forgetting to check the solution for extraneous solutions
Q: Can I use a calculator to solve for x in an equation involving a square root?
A: Yes, you can use a calculator to solve for x in an equation involving a square root. However, it's essential to check the solution for extraneous solutions to ensure that it's valid.
Q: How do I know if the equation 1 = 3 + โ(2x + 7) has a solution?
A: To determine if the equation 1 = 3 + โ(2x + 7) has a solution, you need to check if the expression inside the square root is non-negative. If the expression inside the square root is non-negative, then the equation has a solution. If it's not non-negative, then the equation does not have a solution.
Q: Can I use the quadratic formula to solve for x in the equation 1 = 3 + โ(2x + 7)?
A: No, you cannot use the quadratic formula to solve for x in the equation 1 = 3 + โ(2x + 7). The quadratic formula is used to solve quadratic equations, but the equation 1 = 3 + โ(2x + 7) is not a quadratic equation.
Q: How do I graph the equation 1 = 3 + โ(2x + 7) on a coordinate plane?
A: To graph the equation 1 = 3 + โ(2x + 7) on a coordinate plane, you need to first isolate the square root term. Then, you can graph the square root term as a function of x. The graph of the equation will be a curve that represents the relationship between x and the square root term.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to model real-world phenomena?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to model real-world phenomena. For example, you can use this equation to model the growth of a population or the spread of a disease.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in physics?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in physics, you need to first understand the physical context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in engineering?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in engineering. For example, you can use this equation to model the stress on a material or the flow of a fluid.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in economics?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in economics, you need to first understand the economic context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in computer science?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in computer science. For example, you can use this equation to model the time complexity of an algorithm or the space complexity of a data structure.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in mathematics?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in mathematics, you need to first understand the mathematical context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in statistics?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in statistics. For example, you can use this equation to model the distribution of a random variable or the relationship between two variables.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in data analysis?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in data analysis, you need to first understand the data analysis context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in machine learning?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in machine learning. For example, you can use this equation to model the relationship between the input and output variables in a machine learning model.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in artificial intelligence?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in artificial intelligence, you need to first understand the artificial intelligence context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in computer vision?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in computer vision. For example, you can use this equation to model the relationship between the input and output variables in a computer vision model.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in natural language processing?
A: To use the equation 1 = 3 + โ(2x + 7) to solve a problem in natural language processing, you need to first understand the natural language processing context of the problem. Then, you can use the equation to model the relationship between the variables in the problem. Finally, you can solve for the unknown variable using algebraic techniques.
Q: Can I use the equation 1 = 3 + โ(2x + 7) to solve a problem in robotics?
A: Yes, you can use the equation 1 = 3 + โ(2x + 7) to solve a problem in robotics. For example, you can use this equation to model the relationship between the input and output variables in a robotics model.
Q: How do I use the equation 1 = 3 + โ(2x + 7) to solve a problem in control systems?
A: To use the equation 1 =