91Ó°ÊÓ

Quadratic equations are a type of polynomial equation of degree two, which typically take the form \( ax^2 + bx + c = 0 \). In this expression, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents the unknown variable we need to solve for.
Quadratic equations can have different types of solutions depending on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If negative, the solutions are complex numbers involving imaginary components.

For the given equation \( 9x^2 + 4 = 0 \), there's no linear term (\( bx \)), and the discriminant yields a negative value \( (0 - 4 \, \times \, 9 \, \times \, 4) \), indicating that the solutions involve imaginary numbers.

Imaginary numbers are utilized when dealing with the square roots of negative numbers, offering a solution beyond the scope of real numbers.
The unit imaginary number is denoted as \( i \), where:\[ i = \sqrt{-1} \]
When solving quadratic equations like \( 9x^2 + 4 = 0 \), and upon rearranging to \( x^2 = -\frac{4}{9} \), we encounter the need for imaginary numbers. The use of \( i \) helps us express the solutions as:

• \( x = \pm \frac{2i}{3} \)

This expression shows that each solution is an imaginary number since they include the \( i \) component, indicating a solution on the imaginary axis when plotted on the complex plane.

Complex numbers, often expressed in the form \( a + bi \), can also be represented in polar form. This is particularly useful when visualizing or performing operations on these numbers. The polar form expresses a complex number using its magnitude \( r \) and angle \( \theta \) on the complex plane, given by \( z = r(\cos\theta + i\sin\theta) \). Alternatively, Euler’s formula can be applied: \( z = re^{i\theta} \).To convert \( x = \frac{2i}{3} \) to polar form, notice that:

• The magnitude \( r = \sqrt{(0)^2 + \left(\frac{2}{3}\right)^2} = \frac{2}{3} \)
• The angle \( \theta \) aligns with the positive imaginary axis, meaning \( \theta = \frac{\pi}{2} \)

Thus, the polar form for \( \frac{2i}{3} \) is \( \frac{2}{3}(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}) \), or \( \frac{2}{3}e^{i\frac{\pi}{2}} \). Similarly, for \( -\frac{2i}{3} \), the angle \( \theta = \frac{-\pi}{2} \) would be used.