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

We are given the equation:

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

### Step 1: Simplify the equation
First, let's move all terms to one side of the equation. Add \( 3x \) to both sides to eliminate it from the right side:

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

This simplifies to:

\[
-x^2 = 5
\]

### Step 2: Solve for \( x^2 \)
Now, divide both sides of the equation by \( -1 \):

\[
x^2 = -5
\]

### Step 3: Solve for \( x \)
Take the square root of both sides:

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

Using the imaginary unit \( i \), where \( i = \sqrt{-1} \), we can rewrite \( \sqrt{-5} \) as \( \sqrt{5} \cdot i \).

Thus, the solutions are:

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

### Final Answer:
The solutions are \( x = \sqrt{5}i \) and \( x = -\sqrt{5}i \).

Would you like more details, or do you have any questions about this?

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Here are 5 related questions to expand on the information:

1. What are the properties of complex numbers?
2. How does the square root of a negative number lead to imaginary numbers?
3. Can you solve quadratic equations with real coefficients that have complex solutions?
4. What is the geometric representation of complex numbers on the complex plane?
5. How do complex conjugates relate to solving equations involving imaginary numbers?

**Tip:** Whenever solving quadratic equations, check if the discriminant is negative to identify if the solutions will involve imaginary numbers.

Solve the equation −x^2−3x=5−3x and re-express the answer as a complex number with the imaginary unit.

Learn how to solve the quadratic equation −x^2−3x=5−3x and express the solution in terms of complex numbers. This solution covers key algebraic concepts and the use of imaginary numbers, suitable for students in Grades 10-12.

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