The Imaginary Unit (i) has the following properties. The imaginary number i is defined as the number whose square is -1. That is: Imaginary Numbers are of the form a + bi where b ≠ 0. Complex Numbers are of the form a + bi where a and b are Real Numbers.
We can add and subtract imaginary numbers similar to how we add and subtract terms with variables. Think “like terms.”
Similarly, we can multiply imaginary numbers following the same exponent rules we use for variables.
The absolute value of a complex number is the distance the number lies from the origin in the complex plane. Think Pythagorean Thm..
Larger powers of i can be simplified by dividing the power by 4 and using the remainder to determine the appropriate value.
If z = a + bi is an imaginary number, the its conjugate is z = a – bi. Complex Conjugates can be used to eliminate imaginary numbers from the denominators of fractions. This is very similar to how we rationalize denominators.
Eliminate the Imaginary numbers from the denominator in the following example.
1.When you solve using completing the square on the general formula you get: 2.This is the quadratic formula! 3.Just identify a, b, and c then substitute into the formula.
The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it. An important piece of the quadratic formula is what’s under the radical: b 2 – 4ac This piece is called the discriminant.
The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.
Value of the DiscriminantNature of the Solutions Negative2 imaginary solutions Zero1 Real Solution Positive – perfect square2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational
The graph of a quadratic function is called a parabola. The axis of symmetry is the vertical line that divides the parabola into two identical halves and is written x=a. The vertex (a,b) of the parabola is the point at which the parabola intersects the axis of symmetry and is also a maximum or minimum point of the function.
Given 3 points on the function we can determine the equation of the quadratic.