If $2 \sqrt{x} - \sqrt{6x - 18} = 0$, Then $x = \,?$Show All Of Your Work To Support Your Answer.

Introduction

Radical equations are a type of algebraic equation that involve square roots or other radicals. In this article, we will explore how to solve a radical equation of the form $2 \sqrt{x} - \sqrt{6x - 18} = 0$. Our goal is to isolate the variable x and find its value.

Step 1: Isolate the Radical Expression

The first step in solving a radical equation is to isolate the radical expression on one side of the equation. In this case, we can start by adding $\sqrt{6x - 18}$ to both sides of the equation:

$$2 \sqrt{x} - \sqrt{6x - 18} + \sqrt{6x - 18} = 0 + \sqrt{6x - 18}$$

This simplifies to:

$$2 \sqrt{x} = \sqrt{6x - 18}$$

Step 2: Square Both Sides

The next step is to square both sides of the equation to eliminate the square roots. To do this, we will multiply both sides of the equation by themselves:

$$(2 \sqrt{x})^2 = (\sqrt{6x - 18})^2$$

This simplifies to:

$$4x = 6x - 18$$

Step 3: Simplify the Equation

Now that we have squared both sides of the equation, we can simplify it by combining like terms:

$$4x = 6x - 18$$

Subtracting 4x from both sides gives us:

$$0 = 2x - 18$$

Adding 18 to both sides gives us:

$$18 = 2x$$

Step 4: Solve for x

Finally, we can solve for x by dividing both sides of the equation by 2:

$$x = \frac{18}{2}$$

This simplifies to:

$$x = 9$$

Conclusion

In this article, we have shown how to solve a radical equation of the form $2 \sqrt{x} - \sqrt{6x - 18} = 0$. By following the steps outlined above, we were able to isolate the radical expression, square both sides of the equation, simplify the equation, and finally solve for x. The value of x is 9.

Example Use Cases

Radical equations have many practical applications in mathematics and science. Here are a few examples:

• Geometry: Radical equations can be used to find the length of the sides of a triangle or the area of a circle.
• Physics: Radical equations can be used to model the motion of objects under the influence of gravity or other forces.
• Engineering: Radical equations can be used to design and optimize systems such as bridges, buildings, and electronic circuits.

Tips and Tricks

Here are a few tips and tricks to help you solve radical equations:

• Check your work: Always check your work by plugging your solution back into the original equation.
• Use a calculator: If you are having trouble solving a radical equation, try using a calculator to check your work.
• Simplify the equation: Simplifying the equation can make it easier to solve.

Common Mistakes

Here are a few common mistakes to avoid when solving radical equations:

• Not checking your work: Failing to check your work can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
• Not using a calculator: Failing to use a calculator can make it harder to check your work.

Conclusion

Q: What is a radical equation?

A: A radical equation is a type of algebraic equation that involves square roots or other radicals. It is an equation that contains a variable or expression inside a square root or other radical.

Q: How do I know if an equation is a radical equation?

A: An equation is a radical equation if it contains a variable or expression inside a square root or other radical. For example, the equation $\sqrt{x} + 2 = 5$ is a radical equation because it contains a variable (x) inside a square root.

Q: What are some common types of radical equations?

A: Some common types of radical equations include:

• Simple radical equations: These are equations that contain a single radical expression, such as $\sqrt{x} = 2$.
• Compound radical equations: These are equations that contain multiple radical expressions, such as $\sqrt{x} + \sqrt{y} = 3$.
• Rational radical equations: These are equations that contain a rational expression inside a radical, such as $\sqrt{\frac{x}{y}} = 2$.

Q: How do I solve a radical equation?

A: To solve a radical equation, follow these steps:

1. Isolate the radical expression: Move all terms except the radical expression to the other side of the equation.
2. Square both sides: Square both sides of the equation to eliminate the radical.
3. Simplify the equation: Simplify the equation by combining like terms.
4. Solve for the variable: Solve for the variable by isolating it on one side of 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 checking your work: Failing to check your work can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
• Not using a calculator: Failing to use a calculator can make it harder to check your work.

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

A: To check your work when solving a radical equation, plug your solution back into the original equation and simplify. If the equation is true, then your solution is correct.

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

A: Radical equations have many real-world applications, including:

• Geometry: Radical equations can be used to find the length of the sides of a triangle or the area of a circle.
• Physics: Radical equations can be used to model the motion of objects under the influence of gravity or other forces.
• Engineering: Radical equations can be used to design and optimize systems such as bridges, buildings, and electronic circuits.

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

A: Yes, you can use a calculator to solve radical equations. In fact, calculators can be very helpful when solving radical equations, especially when the equations are complex or involve multiple radical expressions.

Q: What are some tips for solving radical equations?

A: Some tips for solving radical equations include:

• Simplify the equation: Simplifying the equation can make it easier to solve.
• Use a calculator: Using a calculator can make it easier to check your work and simplify the equation.
• Check your work: Always check your work by plugging your solution back into the original equation.

Conclusion

In conclusion, solving radical equations requires a combination of algebraic techniques and attention to detail. By following the steps outlined above and avoiding common mistakes, you can solve radical equations with confidence.