Solve the following quadratic equation by factoring x^2+81=0

To solve the quadratic equation \( x^2 + 81 = 0 \) by factoring, follow these steps:

1. **Rewrite the equation**:  
   The equation is already in the form \( x^2 + 81 = 0 \).

2. **Move the constant to the other side**:  
   \[
   x^2 = -81
   \]

3. **Recognize the difference of squares**:  
   Notice that \( -81 \) can be written as \( -(9^2) \), so:
   \[
   x^2 = -(9^2)
   \]

   This can be written as:
   \[
   x^2 + 9^2 = 0
   \]

   However, this is not factorable in the set of real numbers (since the sum of squares is not factorizable over the reals). To solve this equation, we recognize it as a special case where the solution involves imaginary numbers.

4. **Solve for \(x\)**:  
   Take the square root of both sides:
   \[
   x = \pm \sqrt{-81}
   \]

   Since the square root of a negative number involves an imaginary unit \(i\), where \(i = \sqrt{-1}\), the solution is:
   \[
   x = \pm 9i
   \]

So, the solutions to the equation \( x^2 + 81 = 0 \) are \( x = 9i \) and \( x = -9i \).

Would you like more details or have any questions?

---

Here are some related questions:
1. What is the difference between real and imaginary numbers?
2. How do you factorize a quadratic equation with real solutions?
3. What are the general methods to solve quadratic equations?
4. Can all quadratic equations be solved by factoring?
5. How are imaginary numbers used in practical applications?

**Tip:** Remember that if the discriminant (the expression under the square root in the quadratic formula) is negative, the quadratic equation has imaginary solutions.

Learn how to solve the quadratic equation x^2 + 81 = 0 by factoring, involving imaginary numbers. This problem covers quadratic equations and their solutions using the concept of imaginary numbers. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation