Quadratic Equations & Inequalities

Square Root:

x²=16
√x²=√16
x= ±4
{-4,4}

Factoring

x²=16
x²-16=0
(x-4)(x+4)
x=-4 and x=4
{-4,4}

Quadratic formula

(-b±√(b²-4ac))/(2a)
(-0±√(0²-4(1)(-16)))/(2(1))
(±√64)/2
±8/2
±4
{-4,4}

Standard form: y= ax²+bx+c

Vertex form: y= a(x-h)²+k

Zero-product property: If ab=0, the a=0 or b=0

EX: if (x+3)(x+7)=0, then (x+3)=0 or (x+7)=0

Distance fallen (in feet)at time t (in seconds): d(t)=16t²

Height (in feet)at time t(in seconds): h(t)+h0-16t²

h0 is the object’s initial height in feet

Complex numbers:

i^0=1
i^1=i
i²=-1
i³=-i
i^4= 1

√-a = i√a

EX:

√-5 —> i√5
√-64 —> 8i
√-72 —> 6i√2

When you have stuff like i^a, divide by the power of 4 until you are left with a remainder which will give you the answer

EX:

i^9
9/4 —> 8 +1

* The 8 represents how many times 4 can go into 9, and 1 is the remainder

+1 —> i^1
i^1 —> i

Complex number —> a +bi

*i = √-1

Real number: b=0

imaginary number: b≠ 0

Pure imaginary number: a= 0, b≠0

ALWAYS rewrite √-1 as i

Adding & subtracting

(2+3i)+(4+5i)= 6+8i

add 2+4=6
3i+5i= 8i

(6+7i)- (3+4i)= 3+ 3i

subtract 6-3=3
7i-4i=3i

*tip for subtraction multiply the negative out un till it looks like: 6+7i -3 -4i. Then solve

Solving for x and y

EX: 10x + (3-3y)i = 10 + 9i

Solve for x by pairing the real numbers(or just numbers that don’t have i)

10x=10
x=1

Solve for y by pairing the i’s

(3-3y)i = 9i
(3-3y)=9
-3y=6
y=-2

Multiplying

*use the box method after simplifying for best results

(√2 + 2√-1) (√8 +3√-1)

(√2 +2i)(√8 +3i)
4+3i√2 + 4i√2 +6i²
4+ 7i√2 +6(-1)
-2+ 7i√2

Dividing

*i cannot be left in the denominator

EX:

21-7i/i

21-7i/i multiply by the opposite of i: -i
-21i +7i²/-i²
-21i +7(-1)/-(-1)
-21i -7/ 1
-7 -21i

*if you have something like 1+4i multiply it by 1-4i

*if you have something like 4i√3 multiply by -i√3

Absolute value

EX: |3-4i|

a=3 b=-4
√3²+(-4)²
√9+16
√25
Answer: 5

What to do if it’s to the 5th or 3rd power?

EX:

(2+i)³=2+11i

(2+i)(2+i)(2+i)
(2+i)(2+i)= 3+4i
(3+4i)(2+i)= 2+11i

-√-2

(√2)(√2)(√2)(√2)(√2)
(2)(2)(√2)
-(4√2)(i^5)
-4i√2

Discriminants

Quadratic formula: (-b±√(b²-4ac))/(2a)

Discriminant formula: b²-4ac

*the equations below are used when a or c are not provided to you

b²-4ac>0 and a perfect square —> 2R, rational
- V (touches line twice)
b²-4ac>0 and not a perfect square —> 2R, irrational(still has √)
- V (touches line twice)
b²-4ac=0 —> 1R, rational
- V (touches line once)
b²-4ac<0 —> 2nonR
- V (doesn’t touch line)

Apply to real-world scenarios