Imaginary #s & Completing the Square

Front

What is an imaginary number?

Back

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -1.

Front

What is the value of i^2?

Back

i^2 = -1.

Front

What is the value of i^3?

Back

i^3 = i^2 * i = -1 * i = -i.

Front

What is the value of i^4?

Back

i^4 = (i^2)^2 = (-1)^2 = 1.

Front

How do you simplify i^n for n > 4?

Back

To simplify i^n, divide n by 4 and use the remainder to determine the equivalent power of i: 0 -> 1, 1 -> i, 2 -> -1, 3 -> -i.

Front

What is the process of completing the square?

Back

Completing the square involves rewriting a quadratic equation in the form ax^2 + bx + c into the form a(x-h)^2 + k, where (h,k) is the vertex of the parabola.

Front

What is the vertex form of a quadratic equation?

Back

The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h,k) is the vertex.

Front

How do you complete the square for the equation x^2 + 6x + 5?

Back

1. Take half of b (6), square it (9), and add/subtract it: x^2 + 6x + 9 - 9 + 5 = (x+3)^2 - 4.

Front

What is the result of (3 + i)(7 - 3i)?

Back

(3 + i)(7 - 3i) = 21 - 9i + 7i - 3i^2 = 21 - 2i + 3 = 24 - 2i.

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract the real parts and the imaginary parts separately: (a + bi) - (c + di) = (a - c) + (b - d)i.