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Complex numbers are a type of number that include a real part and an imaginary part. They are written in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
For example, in the complex number \( 3 + 4i \),

• 3 is the real part.
• 4i is the imaginary part.

Imaginary numbers can be tricky because they involve the square root of a negative number, something not possible with real numbers.
The imaginary unit, represented by \( i \), satisfies the equation \( i^2 = -1 \). This means that \( i \) is defined as \( \sqrt{-1} \). Any complex number can be graphed on the complex plane, with the real part displayed on the x-axis and the imaginary part on the y-axis.

Quadratic equations are polynomial equations of degree 2. They have the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
To solve these equations, we often use the quadratic formula, given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides the solutions, or roots, of the quadratic equation. Quadratic equations can have:

• Two real roots if the discriminant (\( b^2 - 4ac \)) is positive.
• One real root if the discriminant is zero.
• Two complex roots if the discriminant is negative.

Solving quadratic equations often involves rearranging the equation to the standard form and then applying the quadratic formula.

The discriminant is a key component in the quadratic formula. It is the part under the square root: \( b^2 - 4ac \). The value of the discriminant determines the nature of the roots of the quadratic equation:

• When the discriminant is positive, \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
• When the discriminant is negative, \( b^2 - 4ac < 0 \), the equation has two complex roots.

For example, in the step-by-step solution above, the discriminant is calculated as:
\[ (4)^2 - 4(3)(2) = 16 - 24 = -8 \]
Since the discriminant is negative, the quadratic equation has two complex roots, incorporating imaginary numbers.

Imaginary numbers are numbers that involve the square root of a negative number. The basic imaginary unit is represented by \( i \), where \( i = \sqrt{-1} \). Using this, we can simplify expressions involving square roots of negative numbers.
For example, \( \sqrt{-8} \) can be written as:
\[ \sqrt{-8} = \sqrt{-1 \cdot 8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i \]
Imaginary numbers are widely used to solve quadratic equations with a negative discriminant. They allow us to find solutions that wouldn't be possible with just real numbers.
In the step-by-step solution, the roots of the quadratic equation were found to include \( \sqrt{-8} \), which was simplified using \( i \) to give:
\[ x = \frac{-2}{3} \pm \frac{\sqrt{2}i}{3} \]
These roots are examples of complex numbers with both real and imaginary parts.