Advanced Level · Self-Study Guide

Algebra 2
& Geometry

The 20 problems students miss most — with memory keys, step-by-step solutions, and instant feedback.

📐 Algebra 2 · 10 Questions 🔷 Geometry · 10 Questions 4 Choices Each Confetti on Correct ✨

Algebra 2 Core Topics

Quadratics · Polynomials · Exponents · Logs · Complex Numbers · Sequences
01
Quadratic Formula⚠ Common Mistake
Solve:  2x² − 5x − 3 = 0
⚡ QUADRATIC FORMULA:   x = (−b ± √(b²−4ac)) / 2a  ·  Identify a, b, c first
A
x = 3 or x = −½
B
x = −3 or x = ½
C
x = 3 or x = −1/2
D
x = 2 or x = −3/2
02
Discriminant
For 3x² + kx + 3 = 0 to have exactly one real solution, what must k equal?
⚡ DISCRIMINANT = b²−4ac  ·  ONE solution → Δ = 0  ·  TWO solutions → Δ > 0  ·  NONE → Δ < 0
A
k = 3
B
k = 6 or k = −6
C
k = 9
D
k = 0
03
Polynomial Division⚠ Tricky
Use the Remainder Theorem: What is the remainder when f(x) = x³ − 4x² + 2x + 5 is divided by (x − 3)?
⚡ REMAINDER THEOREM:   remainder = f(divisor root)  →  divide by (x−c), just compute f(c)
A
−4
B
2
C
5
D
8
04
Logarithms⚠ Sign Trap
Solve:  log₂(x + 3) + log₂(x − 1) = 5
⚡ LOG PRODUCT RULE:   logₐM + logₐN = logₐ(MN)  ·  Always CHECK domain (argument > 0)!
A
x = 4
B
x = −7 or x = 5
C
x = 7
D
x = 5
05
Exponential Equations
Solve:  4x+1 = 8x−1
⚡ SAME BASE TRICK:   rewrite both sides as powers of 2 → set exponents equal
A
x = 3
B
x = 5
C
x = 5
D
x = 7
06
Complex Numbers⚠ i² Trap
Simplify:  (3 + 2i)(1 − 4i)
⚡ FOIL then REPLACE i² = −1  ·  Real part: subtract i² terms · Imaginary part: collect i terms
A
3 − 10i
B
11 − 10i
C
−5 − 10i
D
11 + 2i
07
Rational Exponents
Simplify:  272/3 × 41/2
⚡ RATIONAL EXPONENT: xm/n = (ⁿ√x)m  ·  Denominator = root, Numerator = power
A
18
B
36
C
12
D
24
08
Geometric Sequence
Find the sum of the first 6 terms of a geometric sequence with a₁ = 2 and common ratio r = 3.
⚡ GEO SUM: Sₙ = a₁(1 − rⁿ) / (1 − r)  ·  Use this when r ≠ 1
A
364
B
486
C
244
D
728
09
Vertex Form⚠ Sign Error
The parabola y = 2(x − 3)² + 5 has vertex at which point, and which direction does it open?
⚡ VERTEX FORM: y = a(x − h)² + k  ·  Vertex = (h, k) · a > 0 → opens UP · a < 0 → opens DOWN
A
Vertex (3, 5) · opens upward
B
Vertex (−3, 5) · opens upward
C
Vertex (3, −5) · opens downward
D
Vertex (3, 5) · opens downward
10
Inverse Functions
Find the inverse of:  f(x) = (2x − 4) / 3
⚡ INVERSE: SWAP x and y, then solve for y  ·  f⁻¹ undoes what f does
A
f⁻¹(x) = (3x + 4) / 2
B
f⁻¹(x) = 3x + 4
C
f⁻¹(x) = (3x + 4) / 2
D
f⁻¹(x) = (2x + 4) / 3

Geometry Core Topics

Proofs · 3D Solids · Coordinate Geometry · Transformations · Trigonometry · Circles
11
Volume — Cone
A cone has radius 6 cm and height 9 cm. Find its volume. (Use π ≈ 3.14)
⚡ CONE VOLUME = (1/3)πr²h  ·  Cone = ⅓ of a cylinder with same base and height
A
508.68 cm³
B
339.12 cm³
C
1017.36 cm³
D
226.08 cm³
12
Coordinate Geometry⚠ Midpoint vs Distance
Points A(−2, 3) and B(6, −1). Find the equation of the perpendicular bisector of segment AB.
⚡ PERP BISECTOR:   (1) Find midpoint  (2) Find slope of AB  (3) Perpendicular slope = −1/m
A
y = 2x − 3
B
y = −½x + 1
C
y = 2x + 1
D
y = −2x + 3
13
Right Triangle Trig
In a right triangle, the opposite side = 7 and hypotenuse = 25. Find the adjacent side and cos θ.
⚡ SOH-CAH-TOA  ·  sin=Opp/Hyp · cos=Adj/Hyp · tan=Opp/Adj  ·  Use Pythagoras first!
A
adj = 20, cos θ = 4/5
B
adj = 18, cos θ = 18/25
C
adj = 24, cos θ = 24/25
D
adj = 22, cos θ = 22/25
14
Circle Equation⚠ Sign Trap
What is the center and radius of the circle:  (x + 2)² + (y − 5)² = 49?
⚡ STANDARD CIRCLE: (x−h)² + (y−k)² = r²  ·  Center = (h, k) · WATCH the sign inside the brackets!
A
Center (2, −5), r = 49
B
Center (2, −5), r = 7
C
Center (−2, 5), r = 49
D
Center (−2, 5), r = 7
15
Transformations
Triangle ABC is reflected over the y-axis. If A = (3, −4), what is the image A'?
⚡ REFLECTIONS:   Over y-axis: (x,y)→(−x,y)  ·  Over x-axis: (x,y)→(x,−y)  ·  Only ONE coordinate flips!
A
A' = (−3, −4)
B
A' = (3, 4)
C
A' = (−3, 4)
D
A' = (4, −3)
16
Surface Area — Sphere
Find the surface area of a sphere with radius 5 cm. (π ≈ 3.14)
⚡ SPHERE SURFACE AREA = 4πr²  ·  "Four times pi r squared" — not πr²!
A
157 cm²
B
314 cm²
C
523.33 cm²
D
628 cm²
17
Angle of Elevation⚠ Trig Application
A person stands 40 m from a building. The angle of elevation to the top is 55°. How tall is the building? (tan 55° ≈ 1.428)
⚡ tan(angle) = OPPOSITE / ADJACENT  ·  Height is opposite · Distance from base is adjacent
A
≈ 28.0 m
B
≈ 48.0 m
C
≈ 57.1 m
D
≈ 65.3 m
18
Two-Column Proof Logic
If ∠1 and ∠2 are supplementary and ∠1 = 3x + 10, ∠2 = 5x − 2. Find both angles.
⚡ SUPPLEMENTARY = 180°  ·  Set up: ∠1 + ∠2 = 180 → solve for x → substitute back
A
∠1 = 85°, ∠2 = 95°
B
∠1 = 70°, ∠2 = 110°
C
∠1 = 90°, ∠2 = 90°
D
∠1 = 100°, ∠2 = 80°
19
Dilation
A triangle is dilated with scale factor k = 3, centered at the origin. If one vertex is (2, −4), find the image vertex.
⚡ DILATION from origin: (x, y) → (kx, ky)  ·  Multiply BOTH coordinates by scale factor k
A
(5, −7)
B
(6, −12)
C
(2/3, −4/3)
D
(3, −4)
20
Sector Area⚠ Radians vs Degrees
A circle has radius 10 cm. A sector has central angle 72°. Find the area of the sector. (π ≈ 3.14)
⚡ SECTOR AREA = (θ/360°) × πr²  ·  Fraction of full circle area — same idea as arc length!
A
18.84 cm²
B
157 cm²
C
62.8 cm²
D
31.4 cm²
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Algebra 2
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Geometry