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intro to algebra chapter 12 (complex numbers (graphing (x is real axis…
intro to algebra chapter 12
imaginary numbers
i
square root of -1
\(i^2=-1\)
powers of "I" repeat in groups of 4
\(i^1=i\)
\(i^2=-1\)
\(i^3=-i\)
\(i^4=1\)
if "a" is a real number, then "ai" is imaginary
complex numbers
real part + imaginary part
\(a+bi\)
conjugate
e.g conjugate of \(a+bi\) is \(a-bi\)
multiply the conjugate to get a real number
write real part first
figure out answer by writing that it's equal to \(a+bi\), then solve for "a" and "b"
graphing
x is real axis
plot real part
y is imaginary axis
plot imaginary part
magnitude, distance from origin
real numbers
all numbers which don't have "i"
can be irrational and rational
intro to algebra chapter 13
solving quadratics
completing the square
form a square of binomial
\((a+b)^2=a^2+2ab+b^2\)
look at coefficient of quadratic term if not 1
add stuff to both sides
quadratic formula
\(\frac{-b+-\sqrt{b^{2}-4ac}}{2a}\)
discriminant
\(b^{2}-4ac\)
double root if 0
if roots are imaginary, then they are complex conjugates
can use vieta to help
tips:
remember that something when square rooted, can be negative and positive
put rational number first
simplify by dividing by a common factor of co-efficients
if we square both sides, must check for extraneous solutions