Select All The Correct Answers.What Are The Solutions To This Equation?$\[ 5 - \sqrt[3]{x^2 - 9} = 7 \\]A. 1B. -1C. There Are No Solutions.D. \[$-\sqrt{17}\$\]E. \[$\sqrt{17}\$\]

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Understanding the Equation

The given equation is 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7. To find the solutions to this equation, we need to isolate the variable xx. The equation involves a cube root, which can be challenging to solve. However, with the correct approach, we can find the solutions to this equation.

Isolating the Cube Root

The first step in solving this equation is to isolate the cube root. We can do this by subtracting 5 from both sides of the equation. This gives us:

−x2−93=2-\sqrt[3]{x^2 - 9} = 2

Eliminating the Cube Root

To eliminate the cube root, we need to cube both sides of the equation. This will give us:

(−x2−93)3=23(-\sqrt[3]{x^2 - 9})^3 = 2^3

Simplifying the equation, we get:

−(x2−9)=8-(x^2 - 9) = 8

Solving for x

Now, we can solve for xx by isolating the variable. We can start by multiplying both sides of the equation by -1:

x2−9=−8x^2 - 9 = -8

Adding 9 to both sides of the equation, we get:

x2=1x^2 = 1

Finding the Solutions

To find the solutions to this equation, we need to take the square root of both sides. This gives us:

x=±1x = \pm \sqrt{1}

Simplifying the equation, we get:

x=±1x = \pm 1

Checking the Solutions

Now, we need to check the solutions to see if they satisfy the original equation. We can substitute x=1x = 1 and x=−1x = -1 into the original equation to see if they are true.

For x=1x = 1, we get:

5−12−93=5−−83=5−(−2)=75 - \sqrt[3]{1^2 - 9} = 5 - \sqrt[3]{-8} = 5 - (-2) = 7

This is true, so x=1x = 1 is a solution to the equation.

For x=−1x = -1, we get:

5−(−1)2−93=5−−83=5−(−2)=75 - \sqrt[3]{(-1)^2 - 9} = 5 - \sqrt[3]{-8} = 5 - (-2) = 7

This is also true, so x=−1x = -1 is a solution to the equation.

Conclusion

In conclusion, the solutions to the equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7 are x=1x = 1 and x=−1x = -1. These solutions satisfy the original equation and are therefore the correct answers.

Final Answer

The correct answers are:

  • A. 1
  • B. -1

Discussion

The equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7 is a challenging equation that involves a cube root. However, with the correct approach, we can find the solutions to this equation. The solutions are x=1x = 1 and x=−1x = -1, which satisfy the original equation.

Related Topics

  • Solving equations with cube roots
  • Isolating variables in equations
  • Checking solutions to equations

References

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Q: What is the equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7?

A: The equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7 is a mathematical equation that involves a cube root. The goal is to find the value of xx that satisfies the equation.

Q: How do I solve the equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7?

A: To solve the equation, we need to isolate the cube root by subtracting 5 from both sides of the equation. Then, we can cube both sides of the equation to eliminate the cube root. Finally, we can solve for xx by isolating the variable.

Q: What are the solutions to the equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7?

A: The solutions to the equation 5−x2−93=75 - \sqrt[3]{x^2 - 9} = 7 are x=1x = 1 and x=−1x = -1. These solutions satisfy the original equation and are therefore the correct answers.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, we need to substitute the solution into the original equation and see if it is true. If the solution satisfies the equation, then it is a correct solution.

Q: What if I get a different solution?

A: If you get a different solution, it may be incorrect. However, it's also possible that the solution is correct, but it's not one of the options listed. In this case, you should recheck your work and make sure that you're using the correct steps to solve the equation.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, keep in mind that calculators may not always give you the correct answer, especially if you're using a complex equation. It's always a good idea to double-check your work and make sure that you're using the correct steps to solve the equation.

Q: What if I'm still having trouble solving the equation?

A: If you're still having trouble solving the equation, don't worry! There are many resources available to help you, including online math resources, math textbooks, and even math tutors. Don't be afraid to ask for help if you need it.

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

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

  • Not isolating the cube root
  • Not cubing both sides of the equation
  • Not checking solutions
  • Not using the correct steps to solve the equation

Q: What are some tips for solving equations with cube roots?

A: Some tips for solving equations with cube roots include:

  • Make sure to isolate the cube root
  • Cube both sides of the equation to eliminate the cube root
  • Check solutions to make sure they satisfy the equation
  • Use the correct steps to solve the equation

Related Topics

  • Solving equations with cube roots
  • Isolating variables in equations
  • Checking solutions to equations

References