https://r2.erweima.ai/mathbot/image/1718352130995fdtvzmp3.png

To solve the given problem \(x^2 + 2x = i\), we need to find the complex solutions and determine the product of their real parts.

First, rewrite the equation as:
\[ x^2 + 2x - i = 0 \]

To solve this quadratic equation, we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 2 \), and \( c = -i \).

Substitute these values into the formula:
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-i)}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{4 + 4i}}{2} \]
\[ x = \frac{-2 \pm \sqrt{4(1 + i)}}{2} \]
\[ x = \frac{-2 \pm 2\sqrt{1 + i}}{2} \]
\[ x = -1 \pm \sqrt{1 + i} \]

Now, let's find the square root of \( 1 + i \). To do this, we express \( 1 + i \) in polar form:
\[ 1 + i = \sqrt{2} \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right) \]

Taking the square root:
\[ \sqrt{1 + i} = \sqrt{\sqrt{2} \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right)} \]
\[ \sqrt{1 + i} = \sqrt[4]{2} \left( \cos\frac{\pi}{8} + i \sin\frac{\pi}{8} \right) \]

In rectangular form, this can be approximated as:
\[ \sqrt{1 + i} \approx 1.0987 + 0.4551i \]

Therefore, the solutions are:
\[ x_1 = -1 + (1.0987 + 0.4551i) \approx 0.0987 + 0.4551i \]
\[ x_2 = -1 - (1.0987 + 0.4551i) \approx -2.0987 - 0.4551i \]

The real parts of the solutions are approximately \( 0.0987 \) and \( -2.0987 \).

Finally, the product of their real parts is:
\[ 0.0987 \times -2.0987 \approx -0.2071 \]

Thus, the product of the real parts of the solutions is approximately \(-0.2071\).

Would you like more details or have any questions?

Here are some questions you might want to explore next:
1. How do you solve a quadratic equation with complex coefficients?
2. Can you explain the polar form of a complex number?
3. What is the significance of the roots of a quadratic equation in complex analysis?
4. How can we verify the solutions of a quadratic equation?
5. What are some applications of complex numbers in engineering and physics?

**Tip:** When dealing with complex roots, converting complex numbers to their polar form often simplifies the calculation of roots and powers.

Learn how to solve the quadratic equation x^2 + 2x = i involving complex numbers. Step-by-step solution provided, including the use of the quadratic formula and polar form of complex 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