Solve The Radical Equation 3 X − 11 + 5 = X + 2 \sqrt{3x - 11} + 5 = X + 2 3 X − 11 ​ + 5 = X + 2 .A. X = − 5 X = -5 X = − 5 B. X = 4 X = 4 X = 4 C. X = 4 X = 4 X = 4 Or X = − 5 X = -5 X = − 5 D. X = 4 X = 4 X = 4 Or X = 5 X = 5 X = 5

Introduction

Radical equations are a type of algebraic equation that involves a square root or other radical expression. Solving radical equations requires a combination of algebraic techniques and understanding of the properties of radicals. In this article, we will focus on solving the radical equation $\sqrt{3x - 11} + 5 = x + 2$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Radical Equations

A radical equation is an equation that contains a square root or other radical expression. The general form of a radical equation is:

$$\sqrt{f(x)} = g(x)$$

where $f(x)$ is a function of $x$ and $g(x)$ is another function of $x$. To solve a radical equation, we need to isolate the radical expression and then square both sides of the equation to eliminate the radical.

Step 1: Isolate the Radical Expression

The first step in solving the radical equation $\sqrt{3x - 11} + 5 = x + 2$ is to isolate the radical expression. We can do this by subtracting 5 from both sides of the equation:

$$\sqrt{3x - 11} = x + 2 - 5$$

This simplifies to:

$$\sqrt{3x - 11} = x - 3$$

Step 2: Square Both Sides of the Equation

The next step is to square both sides of the equation to eliminate the radical. We can do this by multiplying both sides of the equation by itself:

$$(\sqrt{3x - 11})^2 = (x - 3)^2$$

This simplifies to:

$$3x - 11 = x^2 - 6x + 9$$

Step 3: Simplify the Equation

The next step is to simplify the equation by combining like terms. We can do this by moving all the terms to one side of the equation:

$$x^2 - 6x + 9 - 3x + 11 = 0$$

This simplifies to:

$$x^2 - 9x + 20 = 0$$

Step 4: Factor the Quadratic Equation

The next step is to factor the quadratic equation. We can do this by finding two numbers whose product is 20 and whose sum is -9. These numbers are -5 and -4, so we can factor the quadratic equation as:

$$(x - 5)(x - 4) = 0$$

Step 5: Solve for x

The final step is to solve for x by setting each factor equal to zero and solving for x. We can do this by setting each factor equal to zero and solving for x:

$$x - 5 = 0 \Rightarrow x = 5$$

$$x - 4 = 0 \Rightarrow x = 4$$

Conclusion

In conclusion, we have solved the radical equation $\sqrt{3x - 11} + 5 = x + 2$ by isolating the radical expression, squaring both sides of the equation, simplifying the equation, factoring the quadratic equation, and solving for x. The solutions to the equation are x = 4 and x = 5.

Answer

The correct answer is D. $x = 4$ or $x = 5$.

Discussion

Radical equations can be challenging to solve, but with the right techniques and understanding of the properties of radicals, they can be solved with ease. In this article, we have demonstrated how to solve a radical equation by isolating the radical expression, squaring both sides of the equation, simplifying the equation, factoring the quadratic equation, and solving for x. We hope that this article has provided a clear and concise explanation of how to solve radical equations and has helped to build your confidence in solving these types of equations.

Additional Tips and Tricks

• When solving radical equations, it is essential to isolate the radical expression before squaring both sides of the equation.
• When squaring both sides of the equation, make sure to square both sides of the equation, not just one side.
• When simplifying the equation, make sure to combine like terms and move all the terms to one side of the equation.
• When factoring the quadratic equation, make sure to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• When solving for x, make sure to set each factor equal to zero and solve for x.

Common Mistakes to Avoid

• Not isolating the radical expression before squaring both sides of the equation.
• Squaring only one side of the equation, not both sides.
• Not combining like terms and moving all the terms to one side of the equation.
• Not factoring the quadratic equation correctly.
• Not setting each factor equal to zero and solving for x.

Conclusion

Introduction

Radical equations can be challenging to solve, but with the right techniques and understanding of the properties of radicals, they can be solved with ease. In this article, we will provide a Q&A guide to help you understand and solve radical equations.

Q: What is a radical equation?

A: A radical equation is an equation that contains a square root or other radical expression. The general form of a radical equation is:

$$\sqrt{f(x)} = g(x)$$

where $f(x)$ is a function of $x$ and $g(x)$ is another function of $x$.

Q: How do I solve a radical equation?

A: To solve a radical equation, you need to isolate the radical expression and then square both sides of the equation to eliminate the radical. Here are the steps:

1. Isolate the radical expression.
2. Square both sides of the equation.
3. Simplify the equation.
4. Factor the quadratic equation.
5. Solve for x.

Q: What is the difference between a radical equation and a quadratic equation?

A: A radical equation is an equation that contains a square root or other radical expression, while a quadratic equation is an equation that contains a squared variable. For example:

Radical equation: $\sqrt{x} = 2$

Quadratic equation: $x^2 = 4$

Q: How do I know if a radical equation has a solution?

A: To determine if a radical equation has a solution, you need to check if the equation is true for all values of x. If the equation is true for all values of x, then the equation has a solution. Otherwise, the equation does not have a solution.

Q: What is the domain of a radical equation?

A: The domain of a radical equation is the set of all values of x for which the equation is true. For example, if the equation is $\sqrt{x} = 2$, then the domain of the equation is $x \geq 0$.

Q: How do I graph a radical equation?

A: To graph a radical equation, you need to plot the graph of the equation on a coordinate plane. You can use a graphing calculator or a computer program to graph the equation.

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

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

• Not isolating the radical expression before squaring both sides of the equation.
• Squaring only one side of the equation, not both sides.
• Not combining like terms and moving all the terms to one side of the equation.
• Not factoring the quadratic equation correctly.
• Not setting each factor equal to zero and solving for x.

Q: How do I check my work when solving a radical equation?

A: To check your work when solving a radical equation, you need to plug your solution back into the original equation and check if it is true. If the equation is true, then your solution is correct. Otherwise, your solution is incorrect.

Conclusion

In conclusion, solving radical equations requires a combination of algebraic techniques and understanding of the properties of radicals. By following the steps outlined in this article and avoiding common mistakes, you can solve radical equations with ease. Remember to isolate the radical expression, square both sides of the equation, simplify the equation, factor the quadratic equation, and solve for x. With practice and patience, you will become proficient in solving radical equations and be able to tackle even the most challenging problems.

Additional Resources

• Online resources: Khan Academy, Mathway, Wolfram Alpha
• Textbooks: Algebra and Trigonometry by Michael Sullivan, College Algebra by James Stewart
• Online communities: Reddit's r/learnmath, Stack Exchange's Mathematics community

Final Tips and Tricks

• Practice, practice, practice: The more you practice solving radical equations, the more comfortable you will become with the techniques and the more confident you will be in your ability to solve them.
• Use online resources: There are many online resources available to help you learn and practice solving radical equations.
• Join online communities: Joining online communities can be a great way to connect with other math enthusiasts and get help when you need it.
• Read textbooks: Reading textbooks can be a great way to learn and practice solving radical equations.