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in an equation
square roots only produce negative and positive numbers when
check for exteraneous
if both sides are squared (or raised to a power divisible by two)
it's positive and negative _______
when you take the square root of both sides of the equation
i=
√-1
SIMPLIFY FIRST
when dealing with imaginary numbers, always
a+bi
complex numbers: formula
a
what is the real part of a+bi
bi
what is the imaginary part of a+bi
6
real part of: 6+7i
0
real part of: 12i
1/2
Cos 60
0
real part of: i/2
2
real part of: 8+3i/4
7i
imaginary part of: 6+7i
12i
imaginary part of: 12i
0 (0i)
imaginary part of: 1/2
i/2 (1/2i)
imaginary part of: i/2
3/2i
imaginary part of: 8+3i/2
a-bi
the complex conjugate of a+bi is
-8
8i^2 simplifies to
a real number
when you multiply a complex number by it's conjugate, you get
denominator of a fraction
you never want to have imaginary numbers on the
when you have imaginary numbers on the denominator of a fraction, you should
multiply the fraction by that thing (or if it's a complex number, by it's conjugate)
f(x)=(x-(2+i))(x-(2-i)) which simplifies to f(x)=x^2-4x+5
when given the roots: 2+i and 2-i, you write the function
another way to get a quadratic expression (a simplified function) when given roots with imaginary numbers
the roots can also be used to get you a, b, and c (which you plug into the quadratic expression to get a simplified function)
-b/a
the sum of the roots (addition) is
c/a
the product of the roots (multiplication) is
can be divided by two
if a number is even it
k
you can tell if a number is divisible by three if
you can tell if a number is divisible by 4 if
the last two numbers of the number are divisible by four (like 1240847903758264)
i^1=i
powers of i
i^1=
i
i^2=
-1
i^3=
-i
i^4=
1