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

Answer 1

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 2

answer in numbeers

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:

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