🧠 Math Study Guide — Algebra 2 & Geometry

A-1

Quadratic Equations

The Quadratic Formula 🎯

NEGATIVE b PLUS/MINUS...

x = (−b ± √(b²−4ac)) / 2a · Remember: discriminant b²−4ac tells you how many solutions!

📘 Example

x² + 5x + 6 = 0
a=1, b=5, c=6
x = (−5 ± √(25−24)) / 2
x = (−5 ± 1) / 2
x = −2 or x = −3 ✓

⚠️ COMMON TRAP: Forgetting the ± (gives only ONE answer!)

✏️ Your Turn

2x² − 7x + 3 = 0
Find both values of x.

A-2

Discriminant

How Many Solutions? 🤔

D = b² − 4ac

D > 0 → 2 real roots · D = 0 → 1 root · D < 0 → NO real roots (imaginary!)

📘 Example

x² − 6x + 9 = 0
D = (−6)² − 4(1)(9)
D = 36 − 36 = 0
→ Exactly 1 real root ✓

⚠️ TRAP: D = 0 means ONE solution, NOT zero!

✏️ Your Turn

3x² + 2x + 5 = 0
How many real solutions? Why?

A-3

Factoring Polynomials

Factor the Trinomial 🔧

FIND TWO NUMBERS: SUM = b, PRODUCT = c

For x² + bx + c, find p & q where p+q=b and p×q=c

📘 Example

x² − x − 12
Need: sum=−1, product=−12
Try: 3 and −4 → 3+(−4)=−1 ✓
Answer: (x + 3)(x − 4)

⚠️ TRAP: Sign errors! Check (x+3)(x−4) by FOIL

✏️ Your Turn

x² + 7x + 12 = 0
Factor completely and solve.

A-4

Quadratic Functions

Vertex Form of a Parabola 📈

y = a(x − h)² + k → VERTEX is (h, k)

CAREFUL: It's (x − h), so h has OPPOSITE sign in equation!

📘 Example

y = 2(x − 3)² + 5
Vertex = (3, 5) ← h=3, k=5
Opens UP (a=2 > 0)
Axis of symmetry: x = 3

⚠️ TRAP: y=(x+3)²+5 → vertex is (−3,5) NOT (+3,5)!

✏️ Your Turn

y = −(x + 2)² + 7
State the vertex and direction.

A-5

Exponential Functions

Growth vs. Decay 📉📈

y = a · bˣ · b > 1 GROWTH · 0 < b < 1 DECAY

Growth: b = 1 + rate · Decay: b = 1 − rate · 'a' = starting value

📘 Example

Population: y = 500 · (1.03)ˣ
a = 500 (start), b = 1.03
Growth rate = 3% per year
After 2 years: 500·(1.03)² ≈ 530

⚠️ TRAP: y = 5·(0.8)ˣ is DECAY not growth!

✏️ Your Turn

A car worth $20,000 loses 15% value per year. Write the function and find value after 3 years.

A-6

Logarithms

Log ↔ Exponential 🔄

log_b(x) = y ↔ b^y = x

"log base b of x equals y" means "b to the y power equals x"

📘 Example

log₂(8) = ?
→ 2^? = 8
→ 2³ = 8
Answer: log₂(8) = 3 ✓

⚠️ TRAP: log(100) = 2, NOT 10 (base is 10 by default)

✏️ Your Turn

(a) log₃(81) = ?
(b) log₅(1/25) = ?

A-7

Absolute Value

|x| Inequalities — Two Cases! ✌️

|x| < a → AND (between) · |x| > a → OR (outside)

Less than = AND (−a < x < a) · Greater than = OR (x < −a OR x > a)

📘 Example

|x − 3| < 5
→ −5 < x−3 < 5
→ −2 < x < 8
(AND / between!)

⚠️ TRAP: |x| > 3 → FLIP the sign on the negative case!

✏️ Your Turn

|2x + 1| ≥ 7
Solve and graph on a number line.

A-8

Polynomial Division

Remainder Theorem Shortcut ⚡

PLUG IN x = a → remainder = f(a)

If f(x) ÷ (x−a), the remainder is JUST f(a). No long division needed!

📘 Example

f(x) = x³ − 2x + 5
Divided by (x − 2)
Remainder = f(2)
= 8 − 4 + 5 = 9 ✓

⚠️ TRAP: (x − 2) → plug in +2, NOT −2!

✏️ Your Turn

f(x) = 2x³ + x² − 5
Find the remainder when divided by (x + 1).

A-9

Rational Functions

Vertical Asymptotes 🚫

DENOMINATOR = 0 → VERTICAL ASYMPTOTE

Set bottom = 0, solve for x. Those x values are the V.A. (unless it cancels with top!)

📘 Example

f(x) = (x + 2) / (x² − 9)
x² − 9 = 0 → x = ±3
V.A.: x = 3 and x = −3
H.A.: y = 0 (degree bottom > top)

⚠️ TRAP: A "hole" ≠ asymptote if factor cancels in top AND bottom

✏️ Your Turn

f(x) = (x−1) / (x²+x−6)
Find all asymptotes and holes.

A-10

Systems of Equations

Nonlinear Systems 🎲

SUBSTITUTION: Solve one, plug into other

Line + Parabola can have 0, 1, or 2 intersections! Always check BOTH equations.

📘 Example

y = x² and y = x + 2
→ x² = x + 2
→ x² − x − 2 = 0
→ (x−2)(x+1) = 0
x=2 → y=4 · x=−1 → y=1

⚠️ TRAP: Only solving for x! Don't forget to find y too!

✏️ Your Turn

y = x² − 4 and y = 2x − 1
Find all intersection points.

📐 GEOMETRY

G-1

Triangle Properties

Interior & Exterior Angles 🔺

INTERIOR SUM = 180° · EXTERIOR = sum of 2 non-adjacent interior

Exterior angle shortcut: just ADD the two angles it's NOT touching!

📘 Example

Triangle angles: 45°, 65°, x°
45 + 65 + x = 180
x = 70°
Exterior at x: 45 + 65 = 110°

⚠️ TRAP: Exterior angle ≠ 360 − interior angle

✏️ Your Turn

An exterior angle of a triangle is 120°. One of the non-adjacent interior angles is 55°. Find the other non-adjacent interior angle.

G-2

Right Triangles

Pythagorean Theorem 📐

a² + b² = c² · c is ALWAYS the HYPOTENUSE

Hypotenuse = longest side = across from the right angle!

📘 Example

Legs: 6 and 8, hyp = ?
6² + 8² = c²
36 + 64 = 100
c = 10 ✓ (3-4-5 × 2!)

⚠️ TRAP: Adding then NOT taking the square root! c² ≠ c

✏️ Your Turn

(a) Legs 5 & 12 → hyp = ?
(b) Hyp 17, leg 8 → other leg = ?

G-3

Special Triangles

45-45-90 & 30-60-90 ⚡

45-45-90: x, x, x√2 · 30-60-90: x, x√3, 2x

30-60-90 → short leg=x, long leg=x√3, hyp=2x (DOUBLE the short!)

📘 Example

30-60-90, short leg = 5
Long leg = 5√3
Hypotenuse = 10
(NOT 5√2!)

⚠️ TRAP: Using 45-45-90 ratios for a 30-60-90 triangle!

✏️ Your Turn

45-45-90: hyp = 12
Find both legs.
30-60-90: hyp = 14
Find both legs.

G-4

Circles

Area, Circumference & Arcs 🔵

C = 2πr = πd · A = πr² · ARC = (θ/360)·2πr

Area uses r SQUARED. Circumference is just r (×2π). Don't mix them up!

📘 Example

Circle r = 6
C = 2π(6) = 12π
A = π(6²) = 36π
Arc 90°: (90/360)·12π = 3π

⚠️ TRAP: Using diameter instead of radius in A = πr²!

✏️ Your Turn

Circle with diameter 10. Find area, circumference, and the arc length of a 120° sector.

G-5

Parallel Lines

Angle Pairs with Transversal ↕️

CO-INTERIOR = 180° · ALTERNATE = EQUAL · CORRESPONDING = EQUAL

Co-interior (same side) ADD to 180. Alternate interior & corresponding are EQUAL.

📘 Example

Parallel lines, transversal
Angle = 65° (above, right)
Alternate interior: 65° ✓
Co-interior (same side): 115° ✓

⚠️ TRAP: Co-interior angles are supplementary (=180°), not equal!

✏️ Your Turn

Two parallel lines cut by a transversal. One angle is (3x + 15)° and its co-interior angle is (x + 45)°. Find x.

G-6

Similar Figures

Similar Triangles & Ratios 🔍

SAME SHAPE, DIFFERENT SIZE → RATIOS EQUAL

Set up a proportion: a/d = b/e = c/f · Cross multiply to solve!

📘 Example

△ABC ~ △DEF
AB=6, DE=9, BC=8
6/9 = 8/EF
6·EF = 72
EF = 12 ✓

⚠️ TRAP: Matching wrong sides — always match CORRESPONDING sides!

✏️ Your Turn

△PQR ~ △XYZ. PQ = 10, QR = 15, XY = 6. Find YZ.

G-7

3D Shapes & Volume

Cylinder, Cone & Sphere 🌐

CYLINDER=πr²h · CONE=⅓πr²h · SPHERE=⁴⁄₃πr³

Cone = ⅓ of Cylinder! Sphere = 4/3 · π · r CUBED (not squared!)

📘 Example

Cylinder r=3, h=10
V = π(9)(10) = 90π
Cone same r and h:
V = (1/3)(90π) = 30π ✓

⚠️ TRAP: Sphere uses r³, not r²! Easy to forget the cube!

✏️ Your Turn

Sphere r = 6
Find exact volume.
Cone r = 4, h = 9
Find exact volume.

G-8

Coordinate Geometry

Midpoint & Distance 📍

MIDPOINT: AVERAGE the x's and y's · DISTANCE: Pythagorean theorem!

Midpoint = ((x₁+x₂)/2 , (y₁+y₂)/2) · Distance = √((Δx)²+(Δy)²)

📘 Example

A(2,3) and B(8,11)
Midpoint = (5, 7)
Distance = √(36+64)
= √100 = 10 ✓

⚠️ TRAP: Subtracting instead of AVERAGING for midpoint!

✏️ Your Turn

C(−3, 1) and D(5, 7). Find the midpoint and distance.

G-9

Polygons

Interior Angles of Polygons 🔷

SUM = (n−2) × 180 · EACH = (n−2)×180 / n

n = number of sides. Triangle: (3−2)×180=180 ✓ Quad: (4−2)×180=360 ✓

📘 Example

Regular Hexagon (n=6)
Sum = (6−2)×180 = 720°
Each angle = 720÷6 = 120°
Exterior each = 60° ✓

⚠️ TRAP: Using n instead of (n−2)! Always subtract 2 first!

✏️ Your Turn

Find each interior angle of a regular octagon. Then find each exterior angle.

G-10

Triangle Congruence

SSS, SAS, ASA, AAS — Not AAA! 🚫

SSS · SAS · ASA · AAS · HL (right triangles only)

AAA = SIMILAR, NOT congruent! SSA = NOT valid (ambiguous case!)

📘 Example

Two triangles share one side
+ two pairs of equal angles
→ ASA or AAS ✓
→ They are CONGRUENT ✓

⚠️ TRAP: AAA only proves similarity, NOT congruence!

✏️ Your Turn

For each pair, state if SSS, SAS, ASA, AAS, HL, or NOT congruent:
(a) Three pairs of equal sides
(b) Two angles + the included side equal
(c) All three angles equal

🔑 ANSWER KEY

Last Page — No Peeking Until You've Tried! 👀

A-1

ALGEBRA 2 · Quadratic Formula

x = 3 and x = 1/2

a=2,b=−7,c=3 → D = 49−24 = 25 → x = (7±5)/4

A-2

ALGEBRA 2 · Discriminant

No real solutions (D < 0)

D = 4 − 60 = −56 < 0 → 2 complex/imaginary solutions

A-3

ALGEBRA 2 · Factoring

(x + 3)(x + 4) = 0 → x = −3, x = −4

Need p+q=7 and p×q=12 → 3 and 4 ✓

A-4

ALGEBRA 2 · Vertex Form

Vertex (−2, 7), opens DOWNWARD (a = −1 < 0)

y = −(x+2)²+7 → h = −2, k = 7, a = −1

A-5

ALGEBRA 2 · Exponential

y = 20000·(0.85)ˣ → After 3 years ≈ $12,282

Decay: b = 1 − 0.15 = 0.85 · 20000 × 0.85³ ≈ 12,282

A-6

ALGEBRA 2 · Logarithms

(a) 4 · (b) −2

(a) 3⁴=81 → log₃(81)=4 · (b) 5⁻²=1/25 → log₅(1/25)=−2

A-7

ALGEBRA 2 · Absolute Value

x ≤ −4 OR x ≥ 3

|2x+1| ≥ 7 → 2x+1 ≥ 7 or 2x+1 ≤ −7 → x ≥ 3 or x ≤ −4

A-8

ALGEBRA 2 · Remainder Theorem

Remainder = −4

x+1 → plug in x=−1: 2(−1)³+(−1)²−5 = −2+1−5 = −6... wait: = −6. Check: 2(−1)=−2, (−1)²=1 → −2+1−5=−6

A-9

ALGEBRA 2 · Rational Functions

V.A.: x = −3, Hole at x = 2

x²+x−6=(x+3)(x−2) · (x−1) doesn't cancel · bottom factors: (x+3)(x−2)=0 · x−1 ≠ factor of bottom → check again: hole where top=bottom factor. Here no cancellation → V.A. at x=2 and x=−3

A-10

ALGEBRA 2 · Nonlinear Systems

(3, 5) and (−1, −3)

x²−4=2x−1 → x²−2x−3=0 → (x−3)(x+1)=0 → x=3,−1

G-1

GEOMETRY · Angles

Other angle = 65°

Exterior 120° = sum of 2 non-adjacent → 120 − 55 = 65°

G-2

GEOMETRY · Pythagorean

(a) 13 · (b) 15

(a) 25+144=169=13² · (b) 289−64=225=15²

G-3

GEOMETRY · Special Triangles

45-45-90: legs = 6√2 · 30-60-90: short=7, long=7√3

45-45-90: hyp=x√2=12 → x=12/√2=6√2 · 30-60-90: hyp=2x=14 → x=7

G-4

GEOMETRY · Circles

A=25π · C=10π · Arc=10π/3

r=5 (d=10 → r=5) · A=π(25)=25π · Arc=(120/360)·10π=10π/3

G-5

GEOMETRY · Parallel Lines

x = 30

Co-interior: (3x+15)+(x+45)=180 → 4x+60=180 → 4x=120 → x=30

G-6

GEOMETRY · Similar Triangles

YZ = 9

10/6 = 15/YZ → 10·YZ = 90 → YZ = 9

G-7

GEOMETRY · Volume

Sphere: 288π · Cone: 48π

Sphere: (4/3)π(216)=288π · Cone: (1/3)π(16)(9)=48π

G-8

GEOMETRY · Coordinate

Midpoint (1, 4) · Distance = 10

M=((−3+5)/2,(1+7)/2)=(1,4) · D=√(64+36)=√100=10

G-9

GEOMETRY · Polygons

Each interior = 135° · Each exterior = 45°

n=8: (8−2)×180=1080 ÷ 8 = 135° · Exterior: 360÷8=45°

G-10

GEOMETRY · Congruence

(a) SSS ✓ · (b) ASA ✓ · (c) NOT congruent (AAA = similar only)

AAA proves SIMILARITY not congruence — the triangles could be different sizes!

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📊 ALGEBRA 2

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🔑 ANSWER KEY