Download notes chapter 5 complex numbers and quadratics

Notes Chapter 5
Complex Numbers and Quadratic Equations
Mr. Durkin
Completing the Square to solve Quadratic Equations with irrational Roots
Steps: 1. Move every term to one side of the equations except
your constant.
2. Create a perfect square out of the Quadratic side of
the equation by adding
equation.
 12 b 
2
to both sides of the
3. Factor the one side of the equations into  x  2 
4. Square root both sides of the equation and solve for
both roots.
b 2
Quadratic Equation - x 
b  b2  4ac
2a
*** Make sure you move the equation so the Quadratic is in
descending order of the powers first. So you’re a, b, and c, are
in the correct place.
2
Discriminant - b  4ac
Roots - Evaluate the Discriminant .
IF: b 2  4ac  0 the roots are Imaginary
b 2  4ac  0 The roots are Real, Rational, Equal
b 2  4ac  0 & Perfect Square the roots are Real, Rat., unequal
b 2  4ac  0 & nonperfect Square the roots are Real, Rat., unequal
Imaginary Numbers -
1  i
When ever you have a square root with a negative
number you can change it to i!!!!!
Powers of i:
i0  1
i1  i
i 2  1
i 3  i
The pattern continues every 4 powers. To find out any powers
higher than 4, divide the power by 4 and find the remainder.
The remainder will tell you which of the values above the
imaginary number matches.
Simplifying imaginary numbers: All of the rules are the same as regular
square roots, except they will have an i in the answer.
Complex Numbers – Any number that can be represented by a + bi form. It
is a whole number and an imaginary number.
Graphing Complex numbers - The y axis becomes Yi axis. The graphs
are done as VECTORS (arrows) from the origin.
Addition/Subtraction of complex numbers - Make sure you distribute
your signs and only add or subtract the whole and the square roots
separately.
Multiply Complex numbers - Multiply and make sure you FOIL the
complex numbers out. Simplify any powers of i.
Dividing Complex Numbers - Put one complex number on top of the other
and rationalize to get the answer.
Additive Identity - 0 + 0i
Multiplicative Identity – 1 + 0i
Additive Inverse - a + bi that can be added to get 0 + 0i.
Multiplicative Inverse – The reciprocal of the complex number,
rationalized.
Sum of the roots -
b
a
Product of the roots -
c
a
Factoring Higher Equations than Quadratics:
1. Look for a common monomial that can be factored out
2. Split and see if you can factor out the same binomial from
each half of the equation. Use the distributive property
to factor the rest of the way.
3. Factor using the difference of two squares!