Math Mastery — Algebra 2 & Geometry

📊 Algebra 2

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Quadratic Functions

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⚡ Memory Point

VERTEX = –b/2a → "Negative B over 2A gets the vertex for FREE"

📝 Worked Example

$ f(x) = 2x^2 - 8x + 3 $
Vertex x-coord: $ x = \dfrac{-(-8)}{2(2)} = \dfrac{8}{4} = 2 $
Then $ f(2) = 2(4) - 16 + 3 = -5 $ → Vertex is $(2, -5)$

Find the vertex of $ f(x) = 3x^2 - 12x + 7 $.
⚠️ Tricky: don't forget to substitute x back!

Use $x = \frac{-b}{2a}$, then plug x into f(x) to get y.

✏️ My work:

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Factoring & Roots

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⚡ Memory Point

SUM–PRODUCT method: Find two numbers whose Sum = b and Product = ac

📝 Worked Example

$2x^2 + 5x + 3$: need sum=5, product=6 → numbers 2,3
$ = (2x+3)(x+1) $

Solve: $ x^2 - 5x - 14 = 0 $
⚠️ Watch the signs! Negative product means one + one −

Find two numbers: sum = −5, product = −14. Try ±2, ±7.

✏️ My work:

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Discriminant & Nature of Roots

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⚡ Memory Point

D = b² − 4ac
D > 0 → TWO real roots | D = 0 → ONE root | D < 0 → NO real roots

📝 Worked Example

$x^2 + 4x + 4 = 0$: $D = 16 - 16 = 0$ → exactly one real root (repeated)
Root: $x = -2$

How many real solutions does $ 2x^2 - 3x + 5 = 0 $ have?
⚠️ Many students forget to multiply 4·a·c correctly!

Calculate $D = (-3)^2 - 4(2)(5)$. Is it positive, zero, or negative?

✏️ My work:

📈

Exponential Functions

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⚡ Memory Point

GROWTH: b>1 | DECAY: 0<b<1
y = a·bˣ → "a = START, b = MULTIPLIER"

📝 Worked Example

$ f(x) = 500 \cdot (1.06)^x $: starts at 500, grows 6% each step.
At x=2: $500 \cdot 1.06^2 = 500 \cdot 1.1236 = 561.8$

A bank account starts with $1000 and grows at 4% per year.
Which function models the balance after $t$ years?
⚠️ Growth rate 4% means multiplier = 1.04, NOT 0.04!

✏️ My work:

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Logarithms

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⚡ Memory Point

log_b(x) = y ↔ b^y = x
"LOG asks the EXPONENT" — log₂(8) = 3 because 2³ = 8

📝 Worked Example

Solve $\log_3(x) = 4$:
Convert: $3^4 = x \Rightarrow x = 81$

Simplify: $ \log_2(32) $
⚠️ Ask yourself: "2 to what power equals 32?"

✏️ My work:

〰️

Rational Expressions

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⚡ Memory Point

CANCEL only FACTORS, never TERMS!
$\dfrac{x+2}{x+5}$ ≠ $\dfrac{2}{5}$ — can't cancel x's that are added!

📝 Worked Example

$\dfrac{x^2-4}{x-2} = \dfrac{(x-2)(x+2)}{x-2} = x+2$ (when $x \neq 2$)

Simplify: $\dfrac{x^2 - 9}{x^2 - x - 6}$
⚠️ Factor BOTH top and bottom first!

✏️ My work:

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Systems of Equations

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⚡ Memory Point

SUBSTITUTION: isolate one variable, then PLUG IN
ELIMINATION: add/subtract equations to kill one variable

📝 Worked Example

$ \begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases} $
Sub y: $3x + (2x+1) = 11 \Rightarrow 5x = 10 \Rightarrow x=2,\ y=5$

Solve: $\begin{cases} y = 3x - 4 \\ 2x + y = 11 \end{cases}$
⚠️ Check your answer by plugging BOTH values back in!

✏️ My work:

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Polynomial Long Division

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⚡ Memory Point

DMSBR → Divide · Multiply · Subtract · Bring down · Repeat

📝 Worked Example

$\dfrac{x^2+5x+6}{x+2}$:
$x^2 \div x = x$ → $x(x+2)=x^2+2x$ → subtract → $3x+6$ → $3x \div x = 3$ → remainder 0
Answer: $x + 3$

Divide: $\dfrac{x^2 + 7x + 10}{x + 2}$
⚠️ After subtracting, don't forget to BRING DOWN the next term!

✏️ My work:

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Sequences & Series

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⚡ Memory Point

Arithmetic: $a_n = a_1 + (n-1)d$ "ADD the difference"
Geometric: $a_n = a_1 \cdot r^{n-1}$ "MULTIPLY the ratio"

📝 Worked Example

Sequence 3, 7, 11, 15… → $d=4$, so $a_n = 3+(n-1)(4) = 4n-1$
10th term: $a_{10} = 4(10)-1 = 39$

Find the 8th term of the arithmetic sequence: $5,\ 9,\ 13,\ 17,\ \ldots$
⚠️ The formula uses (n−1), not n — off-by-one is the #1 mistake!

✏️ My work:

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Complex Numbers

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⚡ Memory Point

i² = −1 | i³ = −i | i⁴ = 1 (cycle of 4!)
To divide complex numbers → MULTIPLY by conjugate $(a-bi)$

📝 Worked Example

$(3+2i)(1-i) = 3 - 3i + 2i - 2i^2 = 3 - i - 2(-1) = 5 - i$

Simplify: $(2 + 3i)(2 - 3i)$
⚠️ This is a "difference of squares" pattern — the imaginary parts cancel!

✏️ My work:

📐 Geometry

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Triangle Similarity

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⚡ Memory Point

AA · SAS · SSS — three ways to prove triangles SIMILAR
"AA = Angle-Angle is the EASIEST proof!"

📝 Worked Example

△ABC ~ △DEF (AA). If AB=6, DE=9, BC=8, find EF.
Ratio = 9/6 = 3/2 → EF = 8 × 3/2 = 12

Two similar triangles have sides in ratio 3:5. If the shorter triangle has a side of 9 cm, what is the corresponding side in the larger triangle?
⚠️ Set up the proportion correctly: $\frac{3}{5} = \frac{9}{x}$

✏️ My work:

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Pythagorean Theorem

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⚡ Memory Point

a² + b² = c² c is always the HYPOTENUSE (longest side, opposite right angle)
Pythagorean triples: 3-4-5, 5-12-13, 8-15-17

📝 Worked Example

Legs = 6 and 8. Find hypotenuse:
$c = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10$

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
⚠️ You MUST know this triple by memory: 5–12–13!

✏️ My work:

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Circle — Arc & Sector

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⚡ Memory Point

Arc length $ = \dfrac{\theta}{360} \cdot 2\pi r$ | Sector area $ = \dfrac{\theta}{360} \cdot \pi r^2$
"Fraction of the FULL circle!"

📝 Worked Example

Circle r=6, central angle=60°.
Arc length $= \dfrac{60}{360} \cdot 2\pi(6) = \dfrac{1}{6} \cdot 12\pi = 2\pi$

A circle has radius 9 cm and a central angle of 120°. What is the area of the sector?
⚠️ Use area formula, not arc length! Don't confuse the two!

✏️ My work:

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Special Right Triangles

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⚡ Memory Point

45-45-90: sides = $x : x : x\sqrt{2}$
30-60-90: sides = $x : x\sqrt{3} : 2x$ "short · short√3 · 2×short"

📝 Worked Example

45-45-90 triangle, leg = 5.
Hypotenuse = $5\sqrt{2}$

In a 30-60-90 triangle, the shorter leg is 7. What is the hypotenuse?
⚠️ Hypotenuse = 2 × shorter leg (NOT the longer leg!)

✏️ My work:

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Parallel Lines & Transversals

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⚡ Memory Point

Co-interior (same-side) angles: ADD to 180°
Alternate interior / Corresponding: EQUAL
"Z-angles = equal · F-angles = equal · C-angles = 180°"

📝 Worked Example

Two parallel lines cut by transversal. Co-interior angle = 110°.
The other co-interior angle = 180° − 110° = 70°

Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is $72°$. What is the other co-interior angle?
⚠️ Co-interior ≠ alternate interior! Co-interior ADDS to 180!

✏️ My work:

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Volume of Solids

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⚡ Memory Point

Cone = ⅓ × Cylinder $V = \frac{1}{3}\pi r^2 h$
Sphere $= \frac{4}{3}\pi r^3$ "Four thirds pi r CUBED"

📝 Worked Example

Cone: r=3, h=8.
$V = \frac{1}{3}\pi(9)(8) = 24\pi \approx 75.4$ cm³

A sphere has radius 3 cm. What is its volume? (Leave in terms of $\pi$)
⚠️ It's r CUBED — not r squared! 3³ = 27, not 9.

✏️ My work:

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Coordinate Geometry

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⚡ Memory Point

Midpoint $= \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$ "AVERAGE of both coordinates"
Distance $= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

📝 Worked Example

Midpoint of $(2,4)$ and $(8,10)$:
$M = \left(\frac{2+8}{2}, \frac{4+10}{2}\right) = (5, 7)$

What is the distance between points $A(1,\ 2)$ and $B(4,\ 6)$?
⚠️ Square the DIFFERENCES, not the coordinates themselves!

✏️ My work:

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Transformations

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⚡ Memory Point

Reflection over x-axis: $(x, y) \to (x, -y)$
Reflection over y-axis: $(x, y) \to (-x, y)$
"Flip over x → y changes sign · Flip over y → x changes sign"

📝 Worked Example

Rotate 90° counterclockwise: $(x,y) \to (-y, x)$
$(3,4) \to (-4,3)$

Point $P(3,\ -5)$ is reflected over the $y$-axis. What are the new coordinates?
⚠️ Only the x-coordinate changes sign when reflecting over the y-axis!

✏️ My work:

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Inscribed Angle Theorem

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⚡ Memory Point

Inscribed angle = HALF the central angle
"Inscribed angle intercepts the SAME arc → half of central"

📝 Worked Example

Central angle = 80° → Inscribed angle = 40°
Inscribed angle = 55° → Intercepted arc = 110°

An inscribed angle intercepts an arc of $140°$. What is the measure of the inscribed angle?
⚠️ Many students confuse which is HALF of which!

✏️ My work:

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Surface Area

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⚡ Memory Point

Cylinder SA = 2πr² + 2πrh
"Two CIRCLES on top & bottom + one RECTANGLE wrapped around" (2πr × h)

📝 Worked Example

Cylinder: r=2, h=5.
$SA = 2\pi(4) + 2\pi(2)(5) = 8\pi + 20\pi = 28\pi$ cm²

A cylinder has radius 3 cm and height 7 cm. What is the total surface area? (Leave in terms of $\pi$)
⚠️ Don't forget BOTH circular ends! Many forget to multiply by 2!

✏️ My work: