Self-Study · Algebra 2 & Geometry

Math Practice
20 Essential Problems

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Algebra 2
Quadratics · Functions · Logarithms · Systems · Sequences
A-01 Quadratic Formula ⭐ Easy
🔑 Key x = (−b ± √(b²−4ac)) / 2a  |  b²−4ac: discriminant
📖 Example
For x² − 5x + 6 = 0, identify a=1, b=−5, c=6.
Discriminant: 25 − 24 = 1 > 0 → two real roots.
x = (5 ± 1) / 2 → x = 3 or x = 2
Solve: 2x² − 7x + 3 = 0
Hint: identify a, b, c first, then plug into the formula.
A-02 Vertex Form ⭐ Easy
🔑 Key f(x) = a(x−h)² + k  →  Vertex = (h, k)
📖 Example
f(x) = 2(x−3)² + 1 → vertex is (3, 1).
Watch the sign! (x−h) means h is positive 3, not negative.
The function f(x) = −3(x + 2)² − 5 has its vertex at which point?
⚠️ Common mistake: confusing the sign of h.
A-03 Discriminant ⭐⭐ Medium
🔑 Key Δ > 0 → 2 real roots  |  Δ = 0 → 1 real root  |  Δ < 0 → no real roots
📖 Example
For x² + 4x + 4 = 0: Δ = 16 − 16 = 0 → exactly one real root (double root).
The parabola just touches the x-axis.
For the equation x² − 6x + k = 0 to have no real solutions, what must be true about k?
A-04 Logarithm Rules ⭐ Easy
🔑 Key log(AB) = logA + logB  |  log(A/B) = logA − logB  |  log(Aⁿ) = n·logA
📖 Example
log₂(8·4) = log₂8 + log₂4 = 3 + 2 = 5
Check: log₂(32) = 5 ✓
Simplify: log₃(27x²), given log₃(x) = p.
A-05 Exponential Equations ⭐ Easy
🔑 Key Same base → set exponents equal  |  aˣ = aʸ ⟹ x = y
📖 Example
4ˣ = 8 → Write both as powers of 2: 2²ˣ = 2³ → 2x = 3 → x = 3/2
Solve for x: 9ˣ = 27
A-06 Arithmetic Sequences ⭐ Easy
🔑 Key aₙ = a₁ + (n−1)d  |  d = common difference
📖 Example
Sequence: 3, 7, 11, 15… → d = 4, a₁ = 3
a₁₀ = 3 + (10−1)×4 = 3 + 36 = 39
In an arithmetic sequence, the first term is 5 and the common difference is −3.
What is the 8th term?
A-07 Geometric Sequences ⭐ Easy
🔑 Key aₙ = a₁ · rⁿ⁻¹  |  r = common ratio
📖 Example
Sequence: 2, 6, 18, 54… → r = 3, a₁ = 2
a₅ = 2 × 3⁴ = 2 × 81 = 162
A geometric sequence has first term 4 and common ratio ½.
What is the 5th term?
A-08 Rational Exponents ⭐⭐ Medium
🔑 Key x^(m/n) = (ⁿ√x)ᵐ  |  "root is n, power is m"
📖 Example
8^(2/3) = (³√8)² = 2² = 4
Always take the root first → smaller numbers are easier!
Evaluate: 27^(4/3)
A-09 Function Composition ⭐⭐ Medium
🔑 Key (f∘g)(x) = f(g(x)) → plug g(x) INTO f(x)
📖 Example
If f(x) = x + 1 and g(x) = 2x,
then (f∘g)(3) = f(g(3)) = f(6) = 7
Let f(x) = x² + 1 and g(x) = 2x − 3.
Find (f ∘ g)(4).
⚠️ Order matters: compute g(4) first, then apply f.
A-10 Inverse Functions ⭐⭐ Medium
🔑 Key To find f⁻¹: swap x and y, then solve for y
📖 Example
f(x) = 3x + 2 → set y = 3x + 2 → swap: x = 3y + 2 → solve: y = (x−2)/3
So f⁻¹(x) = (x−2)/3
Find the inverse of f(x) = 5x − 10.
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Geometry
Triangles · Circles · Similarity · Proofs · Area & Volume
G-01 Pythagorean Theorem ⭐ Easy
🔑 Key a² + b² = c²  |  c = hypotenuse (longest side, opposite right angle)
📖 Example
Legs: 6 and 8 → 6² + 8² = 36 + 64 = 100 → c = √100 = 10
Famous triples: 3-4-5, 5-12-13, 8-15-17
A right triangle has legs of length 9 and 12.
What is the length of the hypotenuse?
G-02 Triangle Angle Sum ⭐ Easy
🔑 Key Interior angles of ANY triangle sum to 180°
📖 Example
Angles 40° and 70° are given → third angle = 180 − 40 − 70 = 70°
In triangle ABC, angle A = (3x + 10)°, angle B = (2x − 5)°, angle C = 75°.
Find the value of x.
G-03 Circle — Arc Length ⭐ Easy
🔑 Key Arc length = (θ/360) × 2πr  |  θ in degrees
📖 Example
Circle radius 6, central angle 60°:
Arc = (60/360) × 2π(6) = (1/6) × 12π = 2π
A circle has radius 9. What is the arc length intercepted by a central angle of 80°?
G-04 Similar Triangles ⭐ Easy
🔑 Key Similar → same angles, proportional sides  |  Set up ratio: a/A = b/B
📖 Example
Triangles with sides 3, 4, 5 and ?, 8, 10 → ratio = 2 → missing side = 6
Two similar triangles have corresponding sides in ratio 3 : 5.
If a side of the smaller triangle is 12, what is the corresponding side of the larger?
G-05 Area of Triangle ⭐ Easy
🔑 Key A = ½ × base × height  |  height must be PERPENDICULAR to base
📖 Example
Base = 10, height = 7 → A = ½ × 10 × 7 = 35
A triangle has a base of 14 cm and a perpendicular height of 9 cm.
What is its area?
G-06 Parallel Lines & Transversal ⭐⭐ Medium
🔑 Key Alternate interior angles = EQUAL  |  Co-interior (same-side) = 180°
📖 Example
Two parallel lines cut by a transversal: alternate interior angles are congruent.
If one is 65°, the alternate interior angle is also 65°.
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures (4x + 15)° and the other measures (2x + 33)°. Find x.
G-07 Volume of Cylinder ⭐ Easy
🔑 Key V = πr²h  |  "area of circle × height"
📖 Example
Cylinder: r = 3, h = 5 → V = π(3²)(5) = 45π
A cylinder has radius 4 cm and height 10 cm.
What is its volume? (Leave answer in terms of π)
G-08 Inscribed Angle Theorem ⭐⭐ Medium
🔑 Key Inscribed angle = ½ × intercepted arc  |  Central angle = intercepted arc
📖 Example
If the intercepted arc is 100°, then the inscribed angle = 50°.
Half the arc!
An inscribed angle in a circle intercepts an arc of 140°.
What is the measure of the inscribed angle?
⚠️ Many students confuse this with central angles.
G-09 30-60-90 Triangle ⭐⭐ Medium
🔑 Key Sides ratio: 1 : √3 : 2  |  short leg : long leg : hypotenuse
📖 Example
Hypotenuse = 10 → short leg = 5, long leg = 5√3
Short leg = 6 → long leg = 6√3, hypotenuse = 12
In a 30-60-90 triangle, the hypotenuse has length 16.
What is the length of the longer leg (opposite 60°)?
G-10 Coordinate Geometry — Midpoint ⭐ Easy
🔑 Key Midpoint = ( (x₁+x₂)/2 , (y₁+y₂)/2 )  |  "average the coordinates"
📖 Example
M of (2, 6) and (8, 2) → M = ((2+8)/2, (6+2)/2) = (5, 4)
Point A is at (−3, 7) and Point B is at (9, −1).
What is the midpoint of segment AB?