Question

Solve the equation −x2−3x=5−3x and re-express the answer as a complex number with the imaginary unit.

Answer 1

answer in numbeers

Answer 2

Let's start by solving the equation:

\[
-x^2 - 3x = 5 - 3x
\]

First, let's move all terms to one side of the equation:

\[
-x^2 - 3x + 3x - 5 = 0
\]

This simplifies to:

\[
-x^2 - 5 = 0
\]

Next, we can multiply the entire equation by -1 to make it easier to work with:

\[
x^2 + 5 = 0
\]

Now, we isolate \(x^2\):

\[
x^2 = -5
\]

Taking the square root of both sides gives us:

\[
x = \pm \sqrt{-5}
\]

This can be expressed using the imaginary unit \(i\):

\[
x = \pm \sqrt{5}i
\]

Thus, the solutions to the equation are:

\[
x = \sqrt{5}i \quad \text{and} \quad x = -\sqrt{5}i
\]

Re-expressing the answer as a complex number with the imaginary unit gives:

\[
x = 0 \pm \sqrt{5}i
\]

So the final answer is:

\[
x = \sqrt{5}i \quad \text{and} \quad x = -\sqrt{5}i
\]

Answer 3

The solutions to the equation \( -x^2 - 3x = 5 - 3x \), when expressed as complex numbers, are:

\[
x = 0 + \sqrt{5}i \quad \text{and} \quad x = 0 - \sqrt{5}i
\]

In numerical form, the two solutions can be expressed as:

\[
x = 0 + 2.236i \quad \text{and} \quad x = 0 - 2.236i
\]

So, in numerical format, the answers are approximately:

\[
x = 2.236i \quad \text{and} \quad x = -2.236i
\]