https://file.aiquickdraw.com/mathbot/file/1731139413352pq1cdunr.jpg

To solve the equation:

\[
z^2 (p^2 + q^2 + 1) = 1
\]

we need to isolate \( z \).

1. **Divide both sides by \( p^2 + q^2 + 1 \):**

   \[
   z^2 = \frac{1}{p^2 + q^2 + 1}
   \]

   Here, \( p^2 + q^2 + 1 \) must not be zero, so \( p^2 + q^2 + 1 \neq 0 \). Since \( p^2 \) and \( q^2 \) are squares of real numbers, \( p^2 + q^2 \geq 0 \), making \( p^2 + q^2 + 1 > 0 \), so this division is valid.

2. **Take the square root of both sides:**

   \[
   z = \pm \sqrt{\frac{1}{p^2 + q^2 + 1}}
   \]

Thus, the solutions for \( z \) are:

\[
z = \pm \frac{1}{\sqrt{p^2 + q^2 + 1}}
\]

This gives us the possible values for \( z \) in terms of \( p \) and \( q \).

Would you like further details or have any questions?

---

Here are five related questions that expand on this topic:

1. How would the solution change if the equation was \( z^3 (p^2 + q^2 + 1) = 1 \)?
2. What are the conditions on \( p \) and \( q \) for the solution to remain real?
3. How would you solve for \( z \) if the equation were \( z^2 (p + q + 1) = 1 \) instead?
4. Can you express \( z \) in polar form if \( p \) and \( q \) were complex numbers?
5. How does this problem relate to solving general quadratic equations in terms of one variable?

**Tip:** When solving equations involving squares, remember to consider both positive and negative roots for a complete solution.

Explore how to solve the equation z²(p² + q² + 1) = 1 with detailed steps. Learn about rational expressions and square root properties. Ideal for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation