Questions: Find the real solutions of the following equation. [ sqrt[11]x^2+2 x=-1 ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Use a comma to separate answers as needed.) B. There are no real solutions.

Find the real solutions of the following equation.
[
sqrt[11]x^2+2 x=-1
]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Use a comma to separate answers as needed.)
B. There are no real solutions.

Transcript text: Find the real solutions of the following equation. \[ \sqrt[11]{x^{2}+2 x}=-1 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \} (Use a comma to separate answers as needed.) B. There are no real solutions.

Solution

Solution Steps

To solve the equation $\sqrt[11]{x^{2}+2 x}=-1$, we need to recognize that the 11th root of a real number is only negative if the number itself is negative. Therefore, we need to solve the equation $x^2 + 2x = -1$ and check if the solutions are real.

Solution Approach

Recognize that $\sqrt[11]{x^{2}+2 x}=-1$ implies $x^2 + 2x = -1$.
Rearrange the equation to $x^2 + 2x + 1 = 0$.
Solve the quadratic equation $x^2 + 2x + 1 = 0$.
Check if the solutions are real.

Step 1: Recognize the Implication of the Equation

Given the equation $\sqrt[11]{x^{2}+2x}=-1$, we recognize that for the 11th root of a number to be $-1$, the number itself must be $-1$. Therefore, we set up the equation: \[ x^2 + 2x = -1 \]

Step 2: Rearrange the Equation

Rearrange the equation to standard quadratic form: \[ x^2 + 2x + 1 = 0 \]

Step 3: Solve the Quadratic Equation

Solve the quadratic equation $x^2 + 2x + 1 = 0$. This can be factored as: \[ (x + 1)^2 = 0 \] Thus, the solution is: \[ x = -1 \]

Step 4: Verify the Solution

Check if the solution is real. Since $x = -1$ is a real number, it is a valid solution.