How Many Real Solutions Does The Equation X − 2 = X + 1 \sqrt{x-2} = X + 1 X − 2 ​ = X + 1 Have?A. 0 B. 1 C. 2 D. Cannot Be Determined From The Graph

Introduction

In mathematics, solving equations is a crucial skill that helps us understand various concepts and relationships between different variables. One type of equation that we often encounter is the radical equation, which involves a square root or other radical expressions. In this article, we will focus on solving the equation $\sqrt{x-2} = x + 1$ and determine the number of real solutions it has.

Understanding the Equation

The given equation is $\sqrt{x-2} = x + 1$. To solve this equation, we need to isolate the variable $x$ and find its possible values. The equation involves a square root, which means that the expression inside the square root must be non-negative. In this case, we have $x-2 \geq 0$, which implies that $x \geq 2$.

Step 1: Square Both Sides

To eliminate the square root, we can square both sides of the equation. This will give us $(\sqrt{x-2})^2 = (x+1)^2$. Simplifying this expression, we get $x-2 = x^2 + 2x + 1$.

Step 2: Rearrange the Equation

Next, we can rearrange the equation to form a quadratic equation. Subtracting $x$ from both sides, we get $-2 = x^2 + x + 1$. Rearranging the terms, we have $x^2 + x + 3 = 0$.

Step 3: Solve the Quadratic Equation

Now, we need to solve the quadratic equation $x^2 + x + 3 = 0$. We can use the quadratic formula to find the solutions: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = 1$, and $c = 3$. Plugging these values into the formula, we get $x = \frac{-1 \pm \sqrt{1 - 4(1)(3)}}{2(1)}$.

Step 4: Simplify the Expression

Simplifying the expression under the square root, we get $x = \frac{-1 \pm \sqrt{-11}}{2}$. Since the square root of a negative number is not a real number, this means that the equation $x^2 + x + 3 = 0$ has no real solutions.

Conclusion

In conclusion, the equation $\sqrt{x-2} = x + 1$ has no real solutions. This is because the quadratic equation $x^2 + x + 3 = 0$ has no real solutions, and the square root of a negative number is not a real number. Therefore, the correct answer is A. 0.

Graphical Representation

To visualize the solution, we can graph the equation $\sqrt{x-2} = x + 1$. The graph will show that the equation has no real solutions, as the two expressions are never equal for any real value of $x$.

Discussion

The equation $\sqrt{x-2} = x + 1$ is a classic example of a radical equation that has no real solutions. This is because the quadratic equation $x^2 + x + 3 = 0$ has no real solutions, and the square root of a negative number is not a real number. The graph of the equation shows that the two expressions are never equal for any real value of $x$, which confirms that the equation has no real solutions.

Real-World Applications

While the equation $\sqrt{x-2} = x + 1$ may seem like a simple algebraic exercise, it has real-world applications in various fields such as physics, engineering, and economics. For example, in physics, the equation can be used to model the motion of an object that is subject to a force that is proportional to the square root of the object's velocity. In engineering, the equation can be used to design systems that involve radical expressions, such as electrical circuits or mechanical systems. In economics, the equation can be used to model the behavior of economic systems that involve radical expressions, such as the behavior of stock prices or interest rates.

Conclusion

Q: What is the main concept behind solving the equation $\sqrt{x-2} = x + 1$?

A: The main concept behind solving the equation $\sqrt{x-2} = x + 1$ is to isolate the variable $x$ and find its possible values. This involves using algebraic techniques such as squaring both sides of the equation and rearranging the terms to form a quadratic equation.

Q: Why is it necessary to square both sides of the equation?

A: Squaring both sides of the equation is necessary to eliminate the square root. This is because the square root of a negative number is not a real number, and we want to find the real solutions of the equation.

Q: What is the quadratic equation that results from squaring both sides of the equation?

A: The quadratic equation that results from squaring both sides of the equation is $x^2 + x + 3 = 0$.

Q: How do we solve the quadratic equation $x^2 + x + 3 = 0$?

A: We can solve the quadratic equation $x^2 + x + 3 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = 1$, and $c = 3$.

Q: What is the result of using the quadratic formula to solve the equation $x^2 + x + 3 = 0$?

A: The result of using the quadratic formula to solve the equation $x^2 + x + 3 = 0$ is $x = \frac{-1 \pm \sqrt{-11}}{2}$. Since the square root of a negative number is not a real number, this means that the equation $x^2 + x + 3 = 0$ has no real solutions.

Q: What is the significance of the graph of the equation $\sqrt{x-2} = x + 1$?

A: The graph of the equation $\sqrt{x-2} = x + 1$ shows that the two expressions are never equal for any real value of $x$. This confirms that the equation has no real solutions.

Q: What are some real-world applications of the equation $\sqrt{x-2} = x + 1$?

A: Some real-world applications of the equation $\sqrt{x-2} = x + 1$ include modeling the motion of an object that is subject to a force that is proportional to the square root of the object's velocity, designing systems that involve radical expressions, and modeling the behavior of economic systems that involve radical expressions.

Q: What is the final answer to the equation $\sqrt{x-2} = x + 1$?

A: The final answer to the equation $\sqrt{x-2} = x + 1$ is that it has no real solutions.

Q: Why is it important to understand the concept of solving radical equations?

A: It is important to understand the concept of solving radical equations because it is a fundamental concept in algebra and has many real-world applications. Radical equations are used to model various phenomena in physics, engineering, and economics, and understanding how to solve them is crucial for making predictions and decisions.

Q: What are some common mistakes to avoid when solving radical equations?

A: Some common mistakes to avoid when solving radical equations include:

• Not checking if the expression inside the square root is non-negative
• Not squaring both sides of the equation
• Not rearranging the terms to form a quadratic equation
• Not using the quadratic formula to solve the quadratic equation
• Not checking if the solutions are real or complex

Q: How can I practice solving radical equations?

A: You can practice solving radical equations by working on exercises and problems that involve radical equations. You can also try graphing the equation and checking if the solutions are real or complex. Additionally, you can try using online resources and tools to help you solve radical equations.