Solve For \[$ C \$\]:$\[ \sqrt{c+11} - \sqrt{c-11} = 3 \\]

Introduction

In this article, we will delve into the world of mathematics and explore a specific equation involving square roots. The equation in question is $\sqrt{c+11} - \sqrt{c-11} = 3$. Our goal is to solve for the variable $c$ and understand the underlying concepts that make this equation work. We will break down the solution into manageable steps, making it easier to grasp for readers who may be struggling with similar problems.

Step 1: Understanding the Equation

The given equation is $\sqrt{c+11} - \sqrt{c-11} = 3$. At first glance, it may seem daunting, but let's take a closer look. We have two square roots, one with a positive term ($c+11$) and the other with a negative term ($c-11$). The difference between these two square roots is equal to $3$. Our task is to isolate the variable $c$ and find its value.

Step 2: Simplifying the Equation

To simplify the equation, we can start by getting rid of the square roots. One way to do this is to multiply both sides of the equation by the conjugate of the left-hand side. The conjugate of $\sqrt{c+11} - \sqrt{c-11}$ is $\sqrt{c+11} + \sqrt{c-11}$. By multiplying both sides by this conjugate, we can eliminate the square roots.

(\sqrt{c+11} - \sqrt{c-11})(\sqrt{c+11} + \sqrt{c-11}) = 3(\sqrt{c+11} + \sqrt{c-11})

Step 3: Expanding and Simplifying

Now that we have multiplied both sides by the conjugate, we can expand and simplify the equation. Using the difference of squares formula, we can rewrite the left-hand side as:

(c+11) - (c-11) = 3(\sqrt{c+11} + \sqrt{c-11})

Simplifying further, we get:

22 = 3(\sqrt{c+11} + \sqrt{c-11})

Step 4: Isolating the Square Roots

Our next step is to isolate the square roots on one side of the equation. We can do this by dividing both sides by $3$:

\frac{22}{3} = \sqrt{c+11} + \sqrt{c-11}

Step 5: Subtracting the Square Roots

Now that we have isolated the square roots, we can subtract the second square root from the first:

\sqrt{c+11} - \sqrt{c-11} = \frac{22}{3} - \sqrt{c-11}

Step 6: Solving for c

We are now ready to solve for $c$. We can start by isolating the square root term on one side of the equation:

\sqrt{c-11} = \frac{22}{3} - \sqrt{c+11}

Next, we can square both sides of the equation to eliminate the square root:

c-11 = \left(\frac{22}{3} - \sqrt{c+11}\right)^2

Expanding and simplifying, we get:

c-11 = \frac{484}{9} - \frac{44}{3}\sqrt{c+11} + c+11

Step 7: Simplifying and Solving

Now that we have simplified the equation, we can solve for $c$. We can start by combining like terms:

c-11 = \frac{484}{9} - \frac{44}{3}\sqrt{c+11} + c+11

Subtracting $c$ from both sides and adding $11$ to both sides, we get:

-11 = \frac{484}{9} - \frac{44}{3}\sqrt{c+11} + 11

Subtracting $11$ from both sides, we get:

-22 = \frac{484}{9} - \frac{44}{3}\sqrt{c+11}

Multiplying both sides by $9$, we get:

-198 = 484 - 132\sqrt{c+11}

Subtracting $484$ from both sides, we get:

-682 = -132\sqrt{c+11}

Dividing both sides by $-132$, we get:

\sqrt{c+11} = \frac{682}{132}

Simplifying, we get:

\sqrt{c+11} = \frac{341}{66}

Squaring both sides, we get:

c+11 = \left(\frac{341}{66}\right)^2

Simplifying, we get:

c+11 = \frac{115601}{4356}

Subtracting $11$ from both sides, we get:

c = \frac{115601}{4356} - 11

Simplifying, we get:

c = \frac{115601 - 48126}{4356}

Simplifying further, we get:

c = \frac{67475}{4356}

Conclusion

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Q: What is the given equation?

A: The given equation is $\sqrt{c+11} - \sqrt{c-11} = 3$. Our goal is to solve for the variable $c$.

Q: Why do we need to multiply both sides by the conjugate?

A: We need to multiply both sides by the conjugate to eliminate the square roots. The conjugate of $\sqrt{c+11} - \sqrt{c-11}$ is $\sqrt{c+11} + \sqrt{c-11}$. By multiplying both sides by this conjugate, we can eliminate the square roots.

Q: What is the difference of squares formula?

A: The difference of squares formula is $(a-b)(a+b) = a^2 - b^2$. We can use this formula to simplify the equation.

Q: How do we isolate the square roots on one side of the equation?

A: We can isolate the square roots on one side of the equation by dividing both sides by $3$. This gives us $\frac{22}{3} = \sqrt{c+11} + \sqrt{c-11}$.

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

A: We need to square both sides of the equation to eliminate the square roots. By squaring both sides, we can get rid of the square roots and solve for $c$.

Q: What is the final solution for c?

A: The final solution for $c$ is $c = \frac{67475}{4356}$.

Q: Can you explain the concept of conjugates in algebra?

A: In algebra, a conjugate is a pair of expressions that, when multiplied together, eliminate the square roots. For example, the conjugate of $\sqrt{a} - \sqrt{b}$ is $\sqrt{a} + \sqrt{b}$. By multiplying both sides of an equation by the conjugate, we can eliminate the square roots and solve for the variable.

Q: How do we know when to use the difference of squares formula?

A: We use the difference of squares formula when we have an equation in the form of $(a-b)(a+b) = a^2 - b^2$. This formula can be used to simplify equations and solve for variables.

Q: Can you provide more examples of solving equations with square roots?

A: Yes, here are a few more examples:

• $\sqrt{x+5} - \sqrt{x-3} = 2$
• $\sqrt{y-2} + \sqrt{y+4} = 5$
• $\sqrt{z+1} - \sqrt{z-6} = 3$

These equations can be solved using the same steps as the original equation.

Q: What are some common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

• Not multiplying both sides by the conjugate
• Not squaring both sides of the equation
• Not simplifying the equation correctly
• Not checking the solution for extraneous solutions

By avoiding these common mistakes, you can ensure that your solutions are accurate and correct.