Questions: Solve the following equation. x^2 + 10 = 0 Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The solution(s) is(are) x= . (Use a comma to separate answers as needed.) B. There are no real solutions.

Solve the following equation.
x^2 + 10 = 0

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

Transcript text: Solve the following equation. \[ x^{2}+10=0 \] Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The solution(s) is(are) $x=$ $\square$ . (Use a comma to separate answers as needed.) B. There are no real solutions.

Solution

Solution Steps

To solve the equation $x^2 + 10 = 0$, we need to find the values of $x$ that satisfy the equation. Since the equation involves a square term, we can isolate $x^2$ and then take the square root of both sides. However, since the right side of the equation will be negative, there are no real solutions. Instead, the solutions will be complex numbers.

Step 1: Identify the Equation

The given equation is $x^2 + 10 = 0$.

Step 2: Rearrange the Equation

Rearrange the equation to isolate the $x^2$ term: \[ x^2 = -10 \]

Step 3: Solve for $x$

To solve for $x$, take the square root of both sides. Since the right side is negative, the solutions will be complex numbers: \[ x = \pm \sqrt{-10} \]

Step 4: Express the Solution in Terms of Imaginary Numbers

The square root of a negative number can be expressed using the imaginary unit $i$, where $i = \sqrt{-1}$: \[ x = \pm \sqrt{10}i \]

Step 5: Calculate the Numerical Values

The numerical value of $\sqrt{10}$ is approximately $3.1623$. Therefore, the solutions are: \[ x = \pm 3.1623i \]