How can I solve the equation (6^x+6^-x)/6 = 2?

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In summary, when solving a hard equation, it is important to carefully consider the type of equation and variables involved in order to determine the most appropriate method to use. Some common mistakes to avoid include forgetting to distribute or combine like terms and making arithmetic errors. To check for a correct solution, it is recommended to plug it back into the original equation, graph the equation, or use a calculator or online tool. If stuck, it can be helpful to take a step back, try a different method, or ask for assistance. The skills learned from solving hard equations can be applied in real-life situations, such as calculating distances or predicting outcomes.
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mathrocks
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I have the given equation: (6^x+6^-x)/6 = 2. How do you solve this?
 
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1. Multiply your equation with [tex]6^{x}[/tex]
2. You have now a quadratic equation in the unknown [tex]y=6^{x}[/tex]
As an alternative, use the identity:
[tex]Cosh(t)=\frac{e^{t}+e^{-t}}{2}[/tex]
 
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To solve this equation, we first need to simplify the left side by using the exponent properties. We can rewrite 6^x as (6^1)^x and 6^-x as (6^(-1))^-x. Using the property (a^m)^n = a^(mn), we can rewrite the equation as (6^(1x)) / 6^(1*(-x)) = 2. This simplifies to 6^(x + (-x)) / 6^(-x) = 2. Now, using the property a^0 = 1, we can simplify further to 6^0 = 6^(-x). This gives us 1 = 6^(-x).

Next, we can take the logarithm of both sides to eliminate the exponent. Using the property log(a^b) = b*log(a), we can rewrite the equation as log(1) = log(6^(-x)). This simplifies to 0 = -x*log(6).

Finally, we can solve for x by dividing both sides by -log(6). This gives us x = 0. Therefore, the solution to the equation is x = 0.
 

1. How do I know which method to use when solving a hard equation?

The method you use to solve an equation depends on the type of equation and the variables involved. Some common methods include substitution, elimination, and graphing. It is important to carefully analyze the equation and determine which method is most appropriate.

2. What are some common mistakes to avoid when solving a hard equation?

Some common mistakes to avoid when solving an equation include forgetting to distribute or combine like terms, making arithmetic errors, and incorrectly applying rules of algebra. It is important to always double check your work and carefully follow each step of the solving process.

3. How can I check my solution to make sure it is correct?

One way to check your solution is to plug it back into the original equation and see if it satisfies the equation. Another method is to graph the equation and see if the solution point falls on the graph. Additionally, you can use a calculator or online tool to verify your solution.

4. What can I do if I get stuck while solving a hard equation?

If you get stuck while solving an equation, take a step back and review the steps you have already completed. It can also be helpful to try a different method or ask a classmate or teacher for assistance. Remember to stay patient and persistent, as solving equations can sometimes be a complex and time-consuming process.

5. How can I apply the skills I learn from solving hard equations in real life?

Many real-life situations involve equations, such as calculating distances, determining interest rates, or predicting future outcomes. By learning how to solve equations, you will develop critical thinking and problem-solving skills that can be applied in various fields, including science, finance, engineering, and more.

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