Homework 6: Radical EquationsCheck For Extraneous Solutions.2. $\sqrt{4k-11} + 15 = 2$

Introduction

Radical equations are a type of algebraic equation that involves a variable within a radical expression. These equations can be challenging to solve, but with a systematic approach, we can find the solutions and check for extraneous solutions. In this article, we will focus on solving radical equations, specifically the equation $\sqrt{4k-11} + 15 = 2$. We will break down the solution process into manageable steps and provide a clear explanation of each step.

Step 1: Isolate the Radical Expression

The first step in solving a radical equation is to isolate the radical expression. In this case, we need to isolate the term $\sqrt{4k-11}$. To do this, we subtract 15 from both sides of the equation:

$$\sqrt{4k-11} + 15 - 15 = 2 - 15$$

This simplifies to:

$$\sqrt{4k-11} = -13$$

Step 2: Square Both Sides

The next step is to square both sides of the equation to eliminate the radical. When we square both sides, we must remember to square the entire expression, not just the radical part:

$$(\sqrt{4k-11})^2 = (-13)^2$$

This simplifies to:

$$4k-11 = 169$$

Step 3: Solve for k

Now that we have a linear equation, we can solve for k:

$$4k-11 = 169$$

Add 11 to both sides:

$$4k = 180$$

Divide both sides by 4:

$$k = 45$$

Step 4: Check for Extraneous Solutions

Before we conclude that k = 45 is the solution to the original equation, we need to check for extraneous solutions. To do this, we substitute k = 45 back into the original equation and check if it is true:

$$\sqrt{4(45)-11} + 15 = 2$$

Simplify the expression inside the square root:

$$\sqrt{179} + 15 = 2$$

Subtract 15 from both sides:

$$\sqrt{179} = -13$$

This is not true, so k = 45 is an extraneous solution.

Conclusion

In this article, we solved the radical equation $\sqrt{4k-11} + 15 = 2$ and checked for extraneous solutions. We found that k = 45 is an extraneous solution, and the original equation has no real solutions. Solving radical equations requires a systematic approach, and it is essential to check for extraneous solutions to ensure that the solution is valid.

Tips and Tricks

• When solving radical equations, it is essential to isolate the radical expression first.
• When squaring both sides, remember to square the entire expression, not just the radical part.
• Always check for extraneous solutions by substituting the solution back into the original equation.

Common Mistakes

• Failing to isolate the radical expression before squaring both sides.
• Squaring only the radical part, not the entire expression.
• Not checking for extraneous solutions.

Real-World Applications

Radical equations have many real-world applications, such as:

• Physics: Radical equations are used to model the motion of objects under the influence of gravity.
• Engineering: Radical equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Radical equations are used in algorithms and data structures to solve problems efficiently.

Practice Problems

1. Solve the radical equation $\sqrt{x-16} + 5 = 3$.
2. Solve the radical equation $\sqrt{2y+5} - 2 = 1$.
3. Solve the radical equation $\sqrt{3z-2} + 1 = 2$.

Conclusion

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Q: What is a radical equation?

A: A radical equation is a type of algebraic equation that involves a variable within a radical expression. Radical expressions are expressions that contain a square root or other root.

Q: How do I solve a radical equation?

A: To solve a radical equation, you need to isolate the radical expression, square both sides, and then solve for the variable. It's essential to check for extraneous solutions by substituting the solution back into the original equation.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid for the original equation. This can happen when you square both sides of the equation and introduce a new solution that is not present in the original equation.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, substitute the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

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

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

• Failing to isolate the radical expression before squaring both sides.
• Squaring only the radical part, not the entire expression.
• Not checking for extraneous solutions.

Q: What are some real-world applications of radical equations?

A: Radical equations have many real-world applications, such as:

• Physics: Radical equations are used to model the motion of objects under the influence of gravity.
• Engineering: Radical equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Radical equations are used in algorithms and data structures to solve problems efficiently.

Q: How do I know if a solution is valid or not?

A: To determine if a solution is valid or not, substitute the solution back into the original equation and check if it is true. If the solution is true, then it is a valid solution.

Q: Can I use a calculator to solve radical equations?

A: Yes, you can use a calculator to solve radical equations. However, it's essential to check the solution by substituting it back into the original equation to ensure that it is valid.

Q: What are some tips for solving radical equations?

A: Some tips for solving radical equations include:

• Isolate the radical expression before squaring both sides.
• Square the entire expression, not just the radical part.
• Check for extraneous solutions by substituting the solution back into the original equation.

Q: Can I use radical equations to solve problems in other areas of mathematics?

A: Yes, radical equations can be used to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Q: How do I know if a radical equation has a solution or not?

A: To determine if a radical equation has a solution or not, try to isolate the radical expression and square both sides. If the resulting equation has a solution, then the original equation has a solution.

Q: Can I use radical equations to model real-world problems?

A: Yes, radical equations can be used to model real-world problems, such as the motion of objects under the influence of gravity or the design of systems, such as bridges and buildings.

Q: How do I choose the correct method for solving a radical equation?

A: To choose the correct method for solving a radical equation, consider the type of equation and the variables involved. If the equation involves a square root, try to isolate the radical expression and square both sides. If the equation involves a higher root, try to isolate the radical expression and use the appropriate formula to solve the equation.

Q: Can I use radical equations to solve problems in other areas of science?

A: Yes, radical equations can be used to solve problems in other areas of science, such as physics, engineering, and computer science.

Q: How do I know if a radical equation has a unique solution or not?

A: To determine if a radical equation has a unique solution or not, try to isolate the radical expression and square both sides. If the resulting equation has a unique solution, then the original equation has a unique solution.