Solve The Equation: $\sqrt{2c+1} = \sqrt{4c}$A. $\frac{3}{2}$B. 2C. $-\frac{1}{2}$D. -2

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Introduction

In this article, we will be solving a simple yet interesting equation involving square roots. The equation is 2c+1=4c\sqrt{2c+1} = \sqrt{4c}, and we will be finding the value of cc that satisfies this equation. We will break down the solution into manageable steps, making it easy to understand and follow along.

Step 1: Square Both Sides of the Equation

The first step in solving this equation is to square both sides of the equation. This will help us eliminate the square roots and make it easier to work with. Squaring both sides of the equation gives us:

(2c+1)2=(4c)2\left(\sqrt{2c+1}\right)^2 = \left(\sqrt{4c}\right)^2

Using the property of exponents that states (ab)c=abc(a^b)^c = a^{bc}, we can simplify the equation to:

2c+1=4c2c+1 = 4c

Step 2: Isolate the Variable

The next step is to isolate the variable cc on one side of the equation. To do this, we can subtract 2c2c from both sides of the equation, which gives us:

1=2c1 = 2c

Step 3: Solve for c

Now that we have isolated the variable cc, we can solve for its value. To do this, we can divide both sides of the equation by 2, which gives us:

c=12c = \frac{1}{2}

Conclusion

In this article, we have solved the equation 2c+1=4c\sqrt{2c+1} = \sqrt{4c} by squaring both sides of the equation and isolating the variable cc. We have found that the value of cc that satisfies this equation is 12\frac{1}{2}.

Answer

The correct answer is:

  • A. 32\frac{3}{2}: This is not the correct answer.
  • B. 2: This is not the correct answer.
  • C. βˆ’12-\frac{1}{2}: This is not the correct answer.
  • D. -2: This is not the correct answer.
  • The correct answer is not listed: The correct answer is 12\frac{1}{2}.

Why is this the correct answer?

This is the correct answer because when we substitute c=12c = \frac{1}{2} into the original equation, we get:

2(12)+1=4(12)\sqrt{2\left(\frac{1}{2}\right)+1} = \sqrt{4\left(\frac{1}{2}\right)}

Simplifying the equation gives us:

1+1=2\sqrt{1+1} = \sqrt{2}

Which is true, so c=12c = \frac{1}{2} is the correct answer.

What if we try the other options?

Let's try substituting the other options into the original equation to see if they work.

  • A. 32\frac{3}{2}: Substituting c=32c = \frac{3}{2} into the original equation gives us:

2(32)+1=4(32)\sqrt{2\left(\frac{3}{2}\right)+1} = \sqrt{4\left(\frac{3}{2}\right)}

Simplifying the equation gives us:

3+1=6\sqrt{3+1} = \sqrt{6}

Which is not true, so c=32c = \frac{3}{2} is not the correct answer.

  • B. 2: Substituting c=2c = 2 into the original equation gives us:

2(2)+1=4(2)\sqrt{2(2)+1} = \sqrt{4(2)}

Simplifying the equation gives us:

5=8\sqrt{5} = \sqrt{8}

Which is not true, so c=2c = 2 is not the correct answer.

  • C. βˆ’12-\frac{1}{2}: Substituting c=βˆ’12c = -\frac{1}{2} into the original equation gives us:

2(βˆ’12)+1=4(βˆ’12)\sqrt{2\left(-\frac{1}{2}\right)+1} = \sqrt{4\left(-\frac{1}{2}\right)}

Simplifying the equation gives us:

βˆ’1+1=βˆ’2\sqrt{-1+1} = \sqrt{-2}

Which is not true, so c=βˆ’12c = -\frac{1}{2} is not the correct answer.

  • D. -2: Substituting c=βˆ’2c = -2 into the original equation gives us:

2(βˆ’2)+1=4(βˆ’2)\sqrt{2(-2)+1} = \sqrt{4(-2)}

Simplifying the equation gives us:

βˆ’4+1=βˆ’8\sqrt{-4+1} = \sqrt{-8}

Which is not true, so c=βˆ’2c = -2 is not the correct answer.

Conclusion

Introduction

In our previous article, we solved the equation 2c+1=4c\sqrt{2c+1} = \sqrt{4c} by squaring both sides of the equation and isolating the variable cc. We found that the value of cc that satisfies this equation is 12\frac{1}{2}. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to square both sides of the equation. This will help us eliminate the square roots and make it easier to work with.

Q: Why do we square both sides of the equation?

A: We square both sides of the equation because it allows us to eliminate the square roots. When we square both sides of the equation, we are essentially getting rid of the square roots and making it easier to work with.

Q: What is the next step after squaring both sides of the equation?

A: The next step after squaring both sides of the equation is to isolate the variable cc on one side of the equation. This can be done by subtracting 2c2c from both sides of the equation.

Q: How do we solve for c?

A: We solve for cc by dividing both sides of the equation by 2. This will give us the value of cc that satisfies the equation.

Q: What if the equation has a negative value inside the square root?

A: If the equation has a negative value inside the square root, we cannot take the square root of a negative number. In this case, we need to use complex numbers to solve the equation.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods to solve the equation. For example, we can use the quadratic formula to solve the equation. However, squaring both sides of the equation is a simple and effective method to solve the equation.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

  • Not squaring both sides of the equation
  • Not isolating the variable cc on one side of the equation
  • Not solving for cc correctly
  • Not checking the solution to make sure it satisfies the original equation

Q: How do we check the solution to make sure it satisfies the original equation?

A: We check the solution by substituting the value of cc back into the original equation and making sure it is true.

Conclusion

In this article, we have answered some frequently asked questions about solving the equation 2c+1=4c\sqrt{2c+1} = \sqrt{4c}. We have covered topics such as squaring both sides of the equation, isolating the variable cc, solving for cc, and checking the solution. By following these steps, we can solve the equation and find the value of cc that satisfies the equation.

Additional Resources

  • For more information on solving equations, check out our article on [Solving Equations](link to article).
  • For more information on square roots, check out our article on [Square Roots](link to article).
  • For more information on complex numbers, check out our article on [Complex Numbers](link to article).

Final Thoughts

Solving the equation 2c+1=4c\sqrt{2c+1} = \sqrt{4c} is a simple and effective way to practice solving equations. By following the steps outlined in this article, we can solve the equation and find the value of cc that satisfies the equation. Remember to always check the solution to make sure it satisfies the original equation.