Solving $7-\sqrt[3]{2-x}=12$: Find X!

Let's dive into solving the equation $7 - \sqrt[3]{2-x} = 12$. Guys, don't worry, we'll break it down step by step so it's super easy to understand. Our main goal here is to isolate $x$, and to do that, we need to get rid of all the other stuff around it. We're gonna use some basic algebra, so buckle up and let's get started!

Step-by-Step Solution

1. Isolate the cube root term: First, we want to isolate the cube root term. We can do this by subtracting 7 from both sides of the equation:

   $7 - \sqrt[3]{2-x} - 7 = 12 - 7$

   This simplifies to:

   $-\sqrt[3]{2-x} = 5$

2. Get rid of the negative sign: Now, let's get rid of the negative sign in front of the cube root. We can multiply both sides of the equation by -1:

   $(-1) * -\sqrt[3]{2-x} = (-1) * 5$

   This gives us:

   $\sqrt[3]{2-x} = -5$

3. Cube both sides: To eliminate the cube root, we'll cube both sides of the equation. This means raising each side to the power of 3:

   $(\sqrt[3]{2-x})^3 = (-5)^3$

   This simplifies to:

   $2-x = -125$

4. Isolate x: Now, let's isolate $x$. We can subtract 2 from both sides of the equation:

   $2 - x - 2 = -125 - 2$

   This gives us:

   $-x = -127$

5. Solve for x: Finally, to solve for $x$, we can multiply both sides of the equation by -1:

   $(-1) * -x = (-1) * -127$

   This results in:

   $x = 127$

So, the solution to the equation $7 - \sqrt[3]{2-x} = 12$ is $x = 127$.

Verification

To make sure our solution is correct, let's plug $x = 127$ back into the original equation and see if it holds true:

$7 - \sqrt[3]{2 - 127} = 12$

$7 - \sqrt[3]{-125} = 12$

Since $\sqrt[3]{-125} = -5$, we have:

$7 - (-5) = 12$

$7 + 5 = 12$

$12 = 12$

The equation holds true, so our solution $x = 127$ is correct!

Additional Tips and Tricks

When solving equations involving radicals, it's always a good idea to check your solution by plugging it back into the original equation. This helps you avoid extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Also, remember to perform operations in the correct order and to pay attention to signs, especially when dealing with negative numbers.

Why This Matters

Understanding how to solve equations like this is super useful in many areas of math and science. Whether you're calculating the trajectory of a rocket, designing a bridge, or just trying to figure out how much pizza to order for your next party, being able to manipulate equations is a valuable skill. Keep practicing, and you'll become a master equation solver in no time!

Common Mistakes to Avoid

• Forgetting to distribute negative signs: When you multiply or divide both sides of an equation by a negative number, make sure to distribute the negative sign to all terms.
• Incorrectly cubing negative numbers: Remember that $(-5)^3 = -5 * -5 * -5 = -125$.
• Not checking for extraneous solutions: Always plug your solution back into the original equation to make sure it's valid.

Conclusion

So there you have it! We've successfully solved the equation $7 - \sqrt[3]{2-x} = 12$ and found that $x = 127$. By following these steps and avoiding common mistakes, you'll be able to tackle similar equations with confidence. Keep practicing and happy solving!

More Practice Problems

Want to test your skills further? Here are a couple more problems you can try:

1. Solve for $x$: $5 + \sqrt[3]{x+1} = 3$
2. Solve for $x$: $10 - \sqrt[3]{3x-5} = 7$

Good luck, and remember, practice makes perfect!

Understanding Cube Roots

Before we wrap up, let's quickly touch on what cube roots are. A cube root of a number is a value that, when multiplied by itself three times, gives you that number. For example, the cube root of 8 is 2 because $2 * 2 * 2 = 8$. Cube roots can be positive, negative, or zero, and every real number has exactly one real cube root.

Advanced Tips

For those who want to go the extra mile, here are some advanced tips for solving equations with cube roots:

• Using substitution: If you encounter a more complex equation, consider using substitution to simplify it. For example, you could let $y = \sqrt[3]{2-x}$, and then solve for $y$ first before solving for $x$.
• Graphing the equation: Another useful technique is to graph the equation and look for the point where it intersects the x-axis. This can give you a visual representation of the solution and help you check your work.

Final Thoughts

Solving equations involving cube roots might seem tricky at first, but with practice and a solid understanding of the basic principles, you can master this skill. Remember to take your time, pay attention to detail, and always check your work. And most importantly, don't be afraid to ask for help if you get stuck. Keep up the great work, and you'll be solving even the most challenging equations in no time! Keep your mind sharp, and keep exploring the fascinating world of mathematics. Who knows, maybe you'll discover a new theorem or solve a problem that no one else has ever been able to figure out. The possibilities are endless!

Importance of Algebra

Algebra is not just a set of rules and symbols; it's a powerful tool for understanding the world around us. From finance to engineering, algebra is used in countless applications. By mastering algebraic concepts, you're not just learning how to solve equations; you're developing critical thinking and problem-solving skills that will serve you well in all areas of life. So embrace the challenge, and enjoy the journey of learning algebra! You've got this, champ!

Next Steps

Now that you've learned how to solve this equation, why not explore other types of equations, such as quadratic equations, exponential equations, or trigonometric equations? Each type presents its own unique challenges and requires different techniques to solve. The more you learn, the more confident you'll become in your mathematical abilities. And who knows, maybe you'll even discover a new passion for mathematics!

Solving Complex Equations

Solving complex equations often involves a combination of techniques, such as substitution, factoring, and using identities. It's important to have a solid understanding of the fundamental principles of algebra before tackling more advanced problems. And remember, practice is key! The more you practice, the better you'll become at recognizing patterns and applying the appropriate techniques.

Conclusion

So there you have it, guys! Solving equations involving cube roots can be a breeze if you follow the right steps and avoid common mistakes. Remember to isolate the cube root, cube both sides of the equation, and always check your solution. With practice, you'll become a pro at solving these types of equations. Now go forth and conquer the world of mathematics!