trig mid

negative discriminatn

no real roots, imaginary

zero discriminant

real, rational, unequal

positive & perf sq.

real, rational, unequal

positive, not perf sq

real, irrational, unequal

how to calculate discriminant

b² - 4ac

sum

-b/a

product

c/a

quadratic inequalities

less than/greater than - OPEN 

or equal to.. - CLOSE

• if x² isnt positive, muktiply by -1, flip sign

undefined fractions

• denom is 0

• state all resrtici=tions, set denom equal to 0, solve fro x

simplifying fractions

factor completely FIRST, then cancel

not identical → cancel this way

(X+3)/(3+x) = 1/1

(x-3)(3-x) = -1 on top

multiplication amongst fractions

factor FIRST, then cancel

Divisional Fractions

flip the divisor (2nd fraction) & multiply, → factor completely, then cancel

like denominators

• add/subtract numerators (turn into one fraction)(KEEP DENOM)

• factor & cancel

Diff Denoms

• find common denom

• write equivalent frac

• add/subtract numerators (one fraction, keep denom)

• Factor & Cancel

Complex Fractions - fractions within fractions

circle denoom to find LCD

multiply EACH part by LCD to get one fraction

factor & cancel

Fractional Equations

— find restrictions ( zero on bottom)

circle denom and find LCD

multiply each by LCD - delete all denoms

solve

check w restricitons

Radicals - adding & subtracting

simplify hten add or subtract like radicals

exponents r perfect sqs!

ndkjfn

multiplying/divide

1st mult/div, coefficients, then radicands, then simplify

*put numerator and denominator under seperate radicals when radical over fraction, simplify numerator THEN DENOM

in these caseds.. 12- sq root of 8 OVER 2

simplify radcial 1st, then reduce fraction if able to in BOTH parts

rationalizing denoms

cant have radical in denom

monomials- multiply by radical in denom, simplify

binomial - multiply by conjugate, then simplify

solving radical equations - algebraically

• isolate radical

• sq both sides

• solve for x

• check answers

imaginary numbers

i^1 - i

i² - -1

i³ - -i

i^4 - 1

cycle of 4 numbers - divide and leftover is the power

i outside radical

add or subtacting imaginary #/complex roots

simplify 1st, add/subtract

multiplying/dividing imaginary #

multiply/divide FIRST, simplify

operations w neg radicals

get radical in terms of i then continue

rationalize denom in imaginary numberes

can never have i or sq root in denominator → js the radical

Complex Numbers

both real and imaginary

• a + bi FORM

+/- complex numbers

+ / - real parts then imaginary parts

multiplying complex numbers

distribute/foil

• put out i

• distribue

• simplify

dividing complex numbers

• multiply by conjugate ( no i or sq root in denom)

standard form of given ± roots,

SUM AND PRODUCT to find equation

multiplicative inverse…:

just the reciprocal

“ find a and b”

wtire two seperate equations for real and imaginary #s

EX: (1+bi) - (a - 5i) = 4 + 3i

→ 1 - a = 4 & b + 5 = 3

• then simplify -