This homework assignment delves into the fascinating world of pure imaginary numbers, a crucial component of understanding and solving quadratic equations. We'll explore their properties, how they arise in solutions, and how to confidently work with them. Let's dive in!
What are Pure Imaginary Numbers?
A pure imaginary number is a complex number whose real part is zero. It's simply a real number multiplied by the imaginary unit, i. Remember that i is defined as the square root of -1 (√-1). Therefore, a pure imaginary number can be expressed in the form bi, where b is a non-zero real number.
Examples of pure imaginary numbers include:
- 2i
- -5i
- √3i
- πi
Working with Pure Imaginary Numbers
Working with pure imaginary numbers involves applying the same rules of arithmetic as with real numbers, but with the key understanding that i² = -1. This seemingly simple rule has profound consequences in simplifying expressions.
Addition and Subtraction: Combine the real and imaginary parts separately. Since we're dealing with pure imaginary numbers, the real part is always zero, so we simply add or subtract the coefficients of i.
Example: (3i) + (7i) = 10i
Multiplication: Multiply the coefficients of i and remember that i² = -1.
Example: (2i) * (5i) = 10i² = 10(-1) = -10
Division: Similar to multiplication, divide the coefficients of i.
Example: (6i) / (3i) = 2
Powers: Remember the cyclical nature of powers of i:
- i¹ = i
- i² = -1
- i³ = -i
- i⁴ = 1
- i⁵ = i (The pattern repeats!)
Pure Imaginary Numbers in Quadratic Equations
Pure imaginary numbers often emerge as solutions to quadratic equations. This happens when the discriminant (the part of the quadratic formula under the square root, b² - 4ac) is negative. Let's recall the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
If b² - 4ac < 0, then the square root yields an imaginary number, leading to complex solutions (which may or may not be pure imaginary). If the real part of the solution is zero, then the solutions are pure imaginary.
Example: Solve the quadratic equation x² + 4 = 0
- Rearrange the equation: x² = -4
- Take the square root of both sides: x = ±√(-4)
- Simplify: x = ±2i
Therefore, the solutions to the quadratic equation x² + 4 = 0 are x = 2i and x = -2i. These are pure imaginary numbers.
Practice Problems
Now it's your turn! Solve the following problems to solidify your understanding of pure imaginary numbers:
- Simplify: (5i) + (-2i) - (8i)
- Multiply: (3i) * (-4i)
- Divide: (12i) / (6i)
- Simplify: i⁷
- Solve the quadratic equation: x² + 9 = 0
- Solve the quadratic equation: 2x² + 8 = 0
This homework should provide you with a comprehensive understanding of pure imaginary numbers and their role in solving quadratic equations. Remember to practice regularly to master this fundamental concept in algebra. Good luck!
