What Is The Solution To 7 X − 4 = 2 X \sqrt{7x-4}=2\sqrt{x} 7 X − 4 ​ = 2 X ​ ?A. − 4 5 -\frac{4}{5} − 5 4 ​ B. 4 3 \frac{4}{3} 3 4 ​ C. 4 5 \frac{4}{5} 5 4 ​ D. − 4 3 -\frac{4}{3} − 3 4 ​

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Introduction

Solving equations involving square roots can be a challenging task, especially when they involve multiple variables. In this article, we will focus on solving the equation $\sqrt{7x-4}=2\sqrt{x}$, which is a classic example of an equation that requires careful manipulation to isolate the variable. We will use algebraic techniques to simplify the equation and find the solution.

Step 1: Square both sides of the equation

To start solving the equation, we need to eliminate the square root on the left-hand side. We can do this by squaring both sides of the equation. This will give us:

$$\left(\sqrt{7x-4}\right)^2 = \left(2\sqrt{x}\right)^2$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the right-hand side of the equation:

$$7x-4 = 4x$$

Step 2: Simplify the equation

Now that we have simplified the equation, we can isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation. We can do this by subtracting $4x$ from both sides of the equation:

$$7x - 4x - 4 = 0$$

This simplifies to:

$$3x - 4 = 0$$

Step 3: Solve for $x$

Now that we have isolated the variable $x$, we can solve for its value. We can do this by adding $4$ to both sides of the equation:

$$3x = 4$$

Next, we can divide both sides of the equation by $3$ to get:

$$x = \frac{4}{3}$$

Conclusion

In this article, we have solved the equation $\sqrt{7x-4}=2\sqrt{x}$ using algebraic techniques. We started by squaring both sides of the equation to eliminate the square root, and then simplified the resulting equation to isolate the variable $x$. Finally, we solved for the value of $x$ by adding $4$ to both sides of the equation and dividing both sides by $3$. The solution to the equation is $x = \frac{4}{3}$.

Answer

The correct answer is B. $\frac{4}{3}$.

Discussion

The equation $\sqrt{7x-4}=2\sqrt{x}$ is a classic example of an equation that requires careful manipulation to isolate the variable. By squaring both sides of the equation and simplifying the resulting equation, we were able to isolate the variable $x$ and solve for its value. This problem requires a good understanding of algebraic techniques, including squaring both sides of an equation and simplifying the resulting equation.

Tips and Tricks

• When solving equations involving square roots, it's often helpful to square both sides of the equation to eliminate the square root.
• When simplifying the resulting equation, be sure to isolate the variable by moving all the terms involving the variable to one side of the equation.
• When solving for the value of the variable, be sure to check your work by plugging the solution back into the original equation.

Related Problems

• $\sqrt{x+3}=2\sqrt{x}$
• $\sqrt{2x-1}=3\sqrt{x}$
• $\sqrt{x-2}=2\sqrt{x-1}$

Conclusion

Solving equations involving square roots can be a challenging task, but with practice and patience, it's possible to develop the skills and techniques needed to solve these types of problems. By following the steps outlined in this article, you should be able to solve the equation $\sqrt{7x-4}=2\sqrt{x}$ and develop a deeper understanding of algebraic techniques.

Introduction

Solving equations involving square roots can be a challenging task, but with practice and patience, it's possible to develop the skills and techniques needed to solve these types of problems. In this article, we will answer some common questions about solving equations involving square roots.

Q: What is the first step in solving an equation involving a square root?

A: The first step in solving an equation involving a square root is to eliminate the square root by squaring both sides of the equation. This will give you a new equation that is easier to work with.

Q: How do I know when to square both sides of an equation?

A: You should square both sides of an equation when the equation involves a square root. Squaring both sides of the equation will eliminate the square root and give you a new equation that is easier to work with.

Q: What is the difference between squaring both sides of an equation and multiplying both sides of an equation by a number?

A: Squaring both sides of an equation involves multiplying both sides of the equation by itself, whereas multiplying both sides of an equation by a number involves multiplying both sides of the equation by a specific number. For example, squaring both sides of the equation $x^2 + 4x = 5$ would give you $x^4 + 8x^3 + 16x^2 = 25$, whereas multiplying both sides of the equation by 2 would give you $2x^2 + 8x = 10$.

Q: How do I simplify an equation after I have squared both sides?

A: After you have squared both sides of an equation, you should simplify the resulting equation by combining like terms and isolating the variable. This will give you a new equation that is easier to work with.

Q: What is the most common mistake people make when solving equations involving square roots?

A: The most common mistake people make when solving equations involving square roots is not checking their work. It's essential to plug the solution back into the original equation to ensure that it is true.

Q: Can I use a calculator to solve equations involving square roots?

A: Yes, you can use a calculator to solve equations involving square roots. However, it's essential to understand the underlying math and be able to solve the equation by hand.

Q: How do I know if I have found the correct solution to an equation involving a square root?

A: You can check your solution by plugging it back into the original equation. If the solution is true, then you have found the correct solution.

Q: What are some common types of equations involving square roots?

A: Some common types of equations involving square roots include:

• $\sqrt{x} = a$
• $\sqrt{x} + b = c$
• $\sqrt{x} - b = c$
• $\sqrt{x} = \sqrt{a} + \sqrt{b}$

Q: How do I solve an equation involving a square root with a variable in the radicand?

A: To solve an equation involving a square root with a variable in the radicand, you should isolate the variable by moving all the terms involving the variable to one side of the equation. Then, you can square both sides of the equation to eliminate the square root.

Q: Can I use the quadratic formula to solve equations involving square roots?

A: Yes, you can use the quadratic formula to solve equations involving square roots. However, it's essential to understand the underlying math and be able to solve the equation by hand.

Q: How do I know if an equation involving a square root has a solution?

A: An equation involving a square root has a solution if the radicand is non-negative and the square root is defined.

Q: What are some real-world applications of solving equations involving square roots?

A: Some real-world applications of solving equations involving square roots include:

• Physics: Solving equations involving square roots is essential in physics to calculate distances, velocities, and accelerations.
• Engineering: Solving equations involving square roots is essential in engineering to calculate stresses, strains, and loads.
• Computer Science: Solving equations involving square roots is essential in computer science to calculate distances, angles, and velocities.

Conclusion

Solving equations involving square roots can be a challenging task, but with practice and patience, it's possible to develop the skills and techniques needed to solve these types of problems. By following the steps outlined in this article, you should be able to answer common questions about solving equations involving square roots and develop a deeper understanding of algebraic techniques.