https://file.aiquickdraw.com/mathbot/file/1727371562790uokd1o83.jpg

The problem asks to find values of \( a \) and \( b \) (with \( a > 0 \)) that satisfy the equation:

\[
(a + bi)^2 = 5 + 12i
\]

### Step-by-Step Solution:

We start by expanding \( (a + bi)^2 \):

\[
(a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2
\]

This results in the equation:

\[
a^2 - b^2 + 2abi = 5 + 12i
\]

Now, we separate the real and imaginary parts:

- Real part: \( a^2 - b^2 = 5 \)
- Imaginary part: \( 2ab = 12 \)

#### Solving the system:

1. From the imaginary part, we solve for \( ab \):

\[
ab = \frac{12}{2} = 6 \quad \text{(Equation 1)}
\]

2. Using Equation 1, we have \( ab = 6 \), so:

\[
b = \frac{6}{a}
\]

Substitute this into the real part \( a^2 - b^2 = 5 \):

\[
a^2 - \left( \frac{6}{a} \right)^2 = 5
\]

Simplifying:

\[
a^2 - \frac{36}{a^2} = 5
\]

Multiply through by \( a^2 \) to eliminate the denominator:

\[
a^4 - 36 = 5a^2
\]

Rearrange into a quadratic form:

\[
a^4 - 5a^2 - 36 = 0
\]

Let \( x = a^2 \), then the equation becomes:

\[
x^2 - 5x - 36 = 0
\]

We solve this quadratic using the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-36)}}{2(1)}
\]

\[
x = \frac{5 \pm \sqrt{25 + 144}}{2} = \frac{5 \pm \sqrt{169}}{2} = \frac{5 \pm 13}{2}
\]

Thus, \( x = \frac{5 + 13}{2} = 9 \) or \( x = \frac{5 - 13}{2} = -4 \) (but \( x = a^2 \geq 0 \), so discard \( -4 \)).

Therefore, \( a^2 = 9 \), so \( a = 3 \) (since \( a > 0 \)).

#### Find \( b \):

Substitute \( a = 3 \) into Equation 1:

\[
ab = 6 \implies 3b = 6 \implies b = 2
\]

### Final Answer:
- \( a = 3 \)
- \( b = 2 \)

Would you like more details or have any questions?

#### Related Questions:
1. What happens if \( a \) were negative? How would the solution change?
2. Can we solve this equation geometrically using the Argand plane?
3. How can this problem be generalized for different right-hand sides, like \( 5 + 8i \)?
4. What is the significance of solving such complex number equations in physics?
5. How does squaring a complex number transform the Argand diagram?

#### Tip:
Always split the real and imaginary parts when solving equations involving complex numbers. This helps simplify the system.

Find the values of a and b (with a > 0) which satisfy (a + bi)^2 = 5 + 12i.

Learn how to solve the complex equation (a + bi)^2 = 5 + 12i by separating real and imaginary parts. The problem involves key algebraic concepts such as the quadratic formula and is suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation