91Ó°ÊÓ

Quadratic equations are polynomial equations of degree two and take the form:

• \( ax^2 + bx + c = 0 \)

Where \( a \), \( b \), and \( c \) are constants. The solutions to these equations are known as the roots and can be found using methods such as factoring, completing the square, or applying the quadratic formula.
Quadratic equations play an important role when working with complex numbers, especially if the roots are given as complex numbers. If one of the roots is a complex number, say \( a + bi \), then its complex conjugate \( a - bi \) is also a root.
This leads to a quadratic equation with real coefficients because the imaginary parts cancel out when the polynomial is expanded. The resulting equation takes the form:

• \( x^2 - 2ax + (a^2 + b^2) = 0 \)

This ensures that although the solutions could be complex, the equation itself maintains real coefficients.

Complex conjugates are a pair of complex numbers that have the same real component and opposite imaginary components.
If \( z = a + bi \) is a complex number, its complex conjugate is \( \bar{z} = a - bi \). The idea of complex conjugates is crucial when solving equations involving complex numbers, as well as in simplifying expressions that include them.
When you multiply a complex number by its conjugate, the result is a real number.
This property is often used to rationalize denominators when dividing complex numbers. For example:

• \( (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \)

This shows how multiplication of conjugates cancels out the imaginary parts, leaving you with a sum of squares, which is a real number.
Using complex conjugates is therefore an essential technique not only for simplifying expressions, but also for solving certain types of quadratic equations.

Finding the inverse of a complex number \( a + bi \) means determining another complex number \( c + di \) such that their product equals 1.
The inverse is particularly useful when performing division with complex numbers. Mathematically, the inverse is:

• \( (a + bi)^{-1} = \frac{1}{a + bi} \)

To simplify this inverse expression, multiply by the complex conjugate of the denominator:

• \( \frac{1}{a + bi} \cdot \frac{a - bi}{a - bi} = \frac{a - bi}{a^2 + b^2} \)

This process cancels out the imaginary part in the denominator, resulting in:

• \( \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i \)

Thus, the inverse of \( a + bi \) is \( c + di \) where \( c = \frac{a}{a^2 + b^2} \) and \( d = -\frac{b}{a^2 + b^2} \).
This understanding is crucial for solving problems in complex arithmetic and for simplifying expressions involving complex numbers.