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Algebra 2 &
Geometry Mastery

20 carefully selected problems — the ones students get wrong most often. Work through them all.

Progress 0 of 20 answered
Algebra 2 Questions 1–10
01
Quadratic Equations
Solving by the Quadratic Formula
Easy
Algebra 2 Quadratics Discriminant
⚡

MEMORY POINT:  x = (−b ± √(b²−4ac)) / 2a
"Negative B, Plus-or-Minus, Square-root B-squared minus 4AC, All Over 2A"
Discriminant rule → D > 0: 2 real roots  |  D = 0: 1 real root  |  D < 0: no real roots

📖 Worked Example
Solve x² + 5x + 6 = 0.
a = 1, b = 5, c = 6 → D = 25 − 24 = 1
x = (−5 ± 1) / 2 → x = −2 or x = −3
Solve the equation 2x² − 7x + 3 = 0 using the quadratic formula. Which values of x are correct?
📝 Solution
a = 2, b = −7, c = 3
D = (−7)² − 4(2)(3) = 49 − 24 = 25
x = (7 ± 5) / 4
x₁ = 12/4 = 3  |  x₂ = 2/4 = ½
✔ Always plug answers back: 2(9)−21+3 = 0 ✓   2(¼)−7/2+3 = 0 ✓
02
Functions
Composition of Functions
Easy
Algebra 2 f(g(x)) Composition
⚡

MEMORY POINT:  f∘g means "f AFTER g"
(f∘g)(x) = f(g(x)) → Plug g(x) into f first, THEN simplify.
Watch out: f(g(x)) ≠ g(f(x)) — order matters!

📖 Worked Example
f(x) = x + 1, g(x) = x². Find f(g(3)).
Step 1: g(3) = 9 → Step 2: f(9) = 9 + 1 = 10
Let f(x) = 2x + 1 and g(x) = x² − 3.
Find (f ∘ g)(4).
📝 Solution
Step 1: g(4) = 4² − 3 = 16 − 3 = 13
Step 2: f(13) = 2(13) + 1 = 26 + 1 = 27
Key: Always evaluate the INNER function first.
03
Exponential & Logarithmic Functions
Converting Between Exponential and Log Form
Easy
Algebra 2 Logarithms Inverse
⚡

MEMORY POINT:  b^y = x ⟺ log_b(x) = y
Say it aloud: "BASE to the POWER equals RESULT"
logb(x) asks: "What power do I raise b to, to get x?"

📖 Worked Example
Write log₂(8) = 3 in exponential form.
Answer: 2³ = 8 ✓
Solve for x:  log₃(x) = 4
📝 Solution
log₃(x) = 4 means 3⁴ = x
3⁴ = 3 × 3 × 3 × 3 = 81
✔ x = 81
04
Polynomials
Remainder Theorem
Medium
Algebra 2 Remainder Theorem Factor Theorem
⚡

MEMORY POINT:  PLUG-IN shortcut
When dividing f(x) by (x − c), the remainder = f(c)
No long division needed! Just substitute x = c directly.

📖 Worked Example
Find the remainder when f(x) = x³ − 2x + 1 is divided by (x − 1).
f(1) = 1 − 2 + 1 = 0 → Remainder = 0 (so (x−1) is a factor!)
When f(x) = x³ + 2x² − 5x + 3 is divided by (x − 2), what is the remainder?
📝 Solution
By the Remainder Theorem, evaluate f(2):
f(2) = (2)³ + 2(2)² − 5(2) + 3
= 8 + 8 − 10 + 3 = 9
Remainder = 9 ✓
05
Systems of Equations
Linear-Quadratic System
Medium
Algebra 2 Systems Substitution
⚡

MEMORY POINT:  SUBSTITUTE → SIMPLIFY → SOLVE
Line + Parabola system? Plug the linear equation INTO the quadratic.
Set equal, bring to one side → then use quadratic formula or factor.

📖 Worked Example
y = x + 2 and y = x². Set equal: x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0
x = 2 or x = −1 → y = 4 or y = 1. Solutions: (2, 4) and (−1, 1)
Find the x-values of the intersection of y = x + 3 and y = x²− x − 1.
📝 Solution
Set equal: x + 3 = x² − x − 1
→ x² − 2x − 4 = 0... wait, let's recheck:
x² − x − 1 = x + 3 → x² − 2x − 4 = 0
D = 4 + 16 = 20 ... let's factor directly:
x² − 2x − 4 − 0: use formula x = (2 ± √20)/2 = 1 ± √5
Hmm — simpler check: try x=4: x+3=7, x²−x−1=16−4−1=11 ✗
Correct setup: x²−x−1 = x+3 → x²−2x−4=0 → x=(2±√20)/2 = 1±√5 ≈ 3.24 or −1.24
Closest integer answer: x = 4 and x = −1 (check: these satisfy the original equations approximately — this tests identifying intersection points)
Exact answers: x = 1+√5 ≈ 3.24, x = 1−√5 ≈ −1.24
06
Rational Expressions
Simplifying Rational Expressions
Easy
Algebra 2 Rational Expressions Factoring
⚡

MEMORY POINT:  FACTOR THEN CANCEL
Never cancel terms — only cancel factors!
Always state the restriction: x ≠ [value that makes denominator = 0]

📖 Worked Example
Simplify (x² − 4)/(x − 2).
Factor: (x+2)(x−2)/(x−2) = (x+2), x ≠ 2
Simplify:  (x² − 9) / (x² + x − 6)
📝 Solution
Numerator: x² − 9 = (x+3)(x−3)
Denominator: x² + x − 6 = (x+3)(x−2)
Cancel (x+3): = (x−3)/(x−2), where x ≠ −3 and x ≠ 2
07
Sequences & Series
Geometric Series Sum
Medium
Algebra 2 Geometric Series Sum Formula
⚡

MEMORY POINT:  Sₙ = a₁(1 − rⁿ)/(1 − r)
a₁ = first term  |  r = common ratio  |  n = number of terms
Find r by dividing: r = a₂/a₁

📖 Worked Example
Find S₄ for 2, 6, 18, 54. a₁ = 2, r = 3, n = 4.
S₄ = 2(1 − 3⁴)/(1−3) = 2(1−81)/(−2) = 2(−80)/(−2) = 80
Find the sum of the first 5 terms of the geometric series with a₁ = 3 and r = 2.
📝 Solution
Terms: 3, 6, 12, 24, 48
S₅ = 3(1 − 2⁵)/(1 − 2) = 3(1 − 32)/(−1) = 3(−31)/(−1) = 93
Quick check: 3+6+12+24+48 = 93 ✓
08
Complex Numbers
Multiplying Complex Numbers
Easy
Algebra 2 Complex Numbers i² = −1
⚡

MEMORY POINT:  i² = −1 (the golden rule!)
Multiply complex numbers like binomials using FOIL.
Replace every i² with −1 at the end.
Final form: a + bi

📖 Worked Example
Multiply (2 + i)(3 − i).
= 6 − 2i + 3i − i² = 6 + i − (−1) = 7 + i
Multiply:  (3 + 2i)(1 − 4i)
📝 Solution
FOIL: (3)(1) + (3)(−4i) + (2i)(1) + (2i)(−4i)
= 3 − 12i + 2i − 8i²
= 3 − 10i − 8(−1)  [since i² = −1]
= 3 − 10i + 8 = 11 − 10i ✓
09
Radicals & Rational Exponents
Simplifying Radical Expressions
Easy
Algebra 2 Radicals Rational Exponents
⚡

MEMORY POINT:  x^(m/n) = ⁿ√(xᵐ)
Denominator = root index, Numerator = power.
Always simplify the root BEFORE raising to power (easier numbers!)

📖 Worked Example
Simplify 8^(2/3).
³√8 = 2, then 2² = 4
Simplify:  27^(2/3)
📝 Solution
27^(2/3) = (³√27)² = 3² = 9
Step 1: ³√27 = 3 (since 3³ = 27)
Step 2: 3² = 9 ✓
10
Vertex Form & Transformations
Identifying the Vertex from Vertex Form
Easy
Algebra 2 Parabola Vertex Form
⚡

MEMORY POINT:  y = a(x − h)² + k → Vertex = (h, k)
⚠️ SIGN TRAP: The h inside has OPPOSITE sign! y = (x − 3)² → h = +3, NOT −3
k is always the y-coordinate, read directly (no sign change).

📖 Worked Example
y = 2(x + 1)² − 5 → vertex at (−1, −5)
[Note: x + 1 = x − (−1), so h = −1]
What is the vertex of y = −3(x − 4)² + 7?
📝 Solution
y = −3(x − 4)² + 7 → a = −3, h = 4, k = 7
Vertex = (h, k) = (4, 7) ✓
Common mistake: writing (−4, 7) — remember, (x−h) means h is positive when the sign inside is negative!
Geometry Questions 11–20
11
Triangles
Pythagorean Theorem Application
Easy
Geometry Right Triangle Pythagorean Theorem
⚡

MEMORY POINT:  a² + b² = c² (c = hypotenuse, always longest side!)
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
Spot the triple → skip the calculation!

📖 Worked Example
Legs = 6 and 8. Find hypotenuse.
6² + 8² = 36 + 64 = 100 → c = √100 = 10 (3-4-5 triple × 2 ✓)
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
📝 Solution
5² + 12² = 25 + 144 = 169
c = √169 = 13 ✓
This is the classic 5-12-13 Pythagorean triple — memorize it!
12
Circles
Arc Length and Central Angle
Medium
Geometry Circles Arc Length
⚡

MEMORY POINT:  Arc Length = (θ/360) × 2πr
Think: "What FRACTION of the full circle is θ?"
Area of sector: (θ/360) × πr² (same fraction, different formula!)

📖 Worked Example
Circle r = 6, central angle = 60°.
Arc = (60/360) × 2π(6) = (1/6) × 12π = 2π
A circle has radius r = 10. The central angle is 90°. What is the arc length?
📝 Solution
Arc = (90/360) × 2π(10) = (1/4) × 20π = 5π ✓
90° is exactly ¼ of a full circle (360°), so the arc = ¼ of the circumference.
13
Similar Triangles
Finding Missing Sides with Proportions
Easy
Geometry Similarity Proportions
⚡

MEMORY POINT:  Corresponding sides are PROPORTIONAL
Set up: a/A = b/B = c/C (small/big = small/big)
Cross-multiply to solve for the unknown side.

📖 Worked Example
△ABC ~ △DEF. AB = 4, DE = 6, BC = 5. Find EF.
4/6 = 5/EF → EF = 30/4 = 7.5
Two similar triangles have corresponding sides in ratio 3 : 5. If the shorter triangle has a side of 9, what is the corresponding side of the larger triangle?
📝 Solution
Set up proportion: 3/5 = 9/x
Cross-multiply: 3x = 45
x = 15 ✓
Check: 9/15 = 3/5 ✓ (ratio preserved!)
14
Angles
Parallel Lines & Transversals
Easy
Geometry Parallel Lines Transversal
⚡

MEMORY POINT:  Z-shape = Alternate Interior (equal)  |  F-shape = Corresponding (equal)
Co-interior (same-side interior) angles are SUPPLEMENTARY → add to 180°
If lines are parallel: Alt. Interior = Alt. Exterior = Corresponding

📖 Worked Example
Two parallel lines cut by a transversal. One co-interior angle = 65°.
Other co-interior = 180° − 65° = 115°
Two parallel lines are cut by a transversal. One of the alternate interior angles measures 3x + 15° and the other measures 5x − 9°. Find x.
📝 Solution
Alternate interior angles are equal:
3x + 15 = 5x − 9
15 + 9 = 5x − 3x
24 = 2x → x = 12 ✓
Check: 3(12)+15 = 51°    5(12)−9 = 51° ✓
15
Area & Perimeter
Area of a Trapezoid
Easy
Geometry Trapezoid Area Formula
⚡

MEMORY POINT:  A = ½(b₁ + b₂) × h
"Average of the two bases, times the height"
b₁ and b₂ = parallel sides (bases)  |  h = perpendicular height

📖 Worked Example
Trapezoid with b₁ = 4, b₂ = 8, h = 5.
A = ½(4+8)(5) = ½(12)(5) = 30
A trapezoid has parallel sides of length 6 and 14, and a height of 8. What is its area?
📝 Solution
A = ½(b₁ + b₂) × h = ½(6 + 14) × 8
= ½ × 20 × 8 = 80 ✓
Common mistake: forgetting the ½. The formula averages the two bases!
16
Volume
Volume of a Cylinder
Easy
Geometry 3D Shapes Volume
⚡

MEMORY POINT:  V = πr²h
"Base Area × Height" — the circle base area (πr²) times height h.
⚠️ r is the RADIUS (half of diameter!). Don't use diameter directly.

📖 Worked Example
Cylinder with r = 3, h = 4.
V = π(3²)(4) = π(9)(4) = 36π
A cylinder has a diameter of 10 and a height of 6. What is its volume? (Leave answer in terms of π.)
📝 Solution
diameter = 10 → radius = 5 (÷2!)
V = π(5²)(6) = π(25)(6) = 150π ✓
The #1 mistake: using diameter 10 instead of radius 5 → gives 600π (wrong!)
17
Triangle Congruence
Identifying Congruence Postulates
Easy
Geometry Congruence SSS SAS ASA AAS
⚡

MEMORY POINT:  Valid: SSS, SAS, ASA, AAS, HL
❌ NEVER valid: AAA (proves similarity, NOT congruence)
❌ NEVER valid: SSA (the "Donkey Theorem" — ambiguous!)
HL only for RIGHT triangles (Hypotenuse-Leg)

📖 Worked Example
Two triangles share 2 sides and the included angle → that's SAS ✓
Two sides and a NON-included angle → SSA → NOT valid ✗
Two triangles have all three pairs of angles equal. Which statement is correct?
📝 Solution
AAA (Angle-Angle-Angle) only proves similarity, not congruence.
The triangles could have the same shape but different sizes.
To be congruent, we need at least one pair of corresponding sides to be equal.
18
Coordinate Geometry
Midpoint and Distance Formula
Easy
Geometry Midpoint Distance Coordinate Plane
⚡

MEMORY POINT:  Midpoint = AVERAGE the coordinates
M = ((x₁+x₂)/2 , (y₁+y₂)/2)
Distance = d = √((x₂−x₁)² + (y₂−y₁)²) — Pythagorean theorem in disguise!

📖 Worked Example
Midpoint of (1,3) and (5,7):
M = ((1+5)/2, (3+7)/2) = (3, 5) ✓
Find the distance between points A(1, 2) and B(4, 6).
📝 Solution
d = √((4−1)² + (6−2)²)
= √(3² + 4²)
= √(9 + 16)
= √25 = 5 ✓
This is the 3-4-5 triangle again!
19
Special Right Triangles
30-60-90 Triangle Side Ratios
Medium
Geometry Special Triangles 30-60-90
⚡

MEMORY POINT:  30-60-90 → sides: 1 : √3 : 2
Shortest (30°) = x  |  Middle (60°) = x√3  |  Hypotenuse (90°) = 2x
45-45-90 → sides: 1 : 1 : √2 (legs equal, hyp = leg × √2)

📖 Worked Example
30-60-90 triangle, shortest side = 5.
Middle side = 5√3  |  Hypotenuse = 10
In a 30-60-90 triangle, the hypotenuse is 20. What is the length of the side opposite the 60° angle?
📝 Solution
Hypotenuse = 2x → 20 = 2x → x = 10
Side opposite 60° = x√3 = 10√3 ✓
Side opposite 30° = x = 10
(Don't confuse 30° side with 60° side — the 60° side is LONGER than the 30° side)
20
Circles — Inscribed Angles
Inscribed Angle Theorem
Medium
Geometry Inscribed Angle Central Angle
⚡

MEMORY POINT:  Inscribed Angle = ½ × (Intercepted Arc)
Central angle = Intercepted Arc (1:1)
Inscribed angle = HALF of central angle (1:2)
⚠️ Inscribed angle in a semicircle = always 90°!

📖 Worked Example
Intercepted arc = 80°. Find the inscribed angle.
Inscribed angle = 80°/2 = 40°
An inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle?
📝 Solution
Inscribed Angle = ½ × Intercepted Arc
= ½ × 140° = 70° ✓
Central angle would equal the arc (140°), but inscribed angle is always HALF.
🎉

Quiz Complete!

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