📐 Math Mastery — Algebra 2 & Geometry

📊 Algebra 2 — 10 Core Questions

Tricky spots marked ⚠️ — don't rush these ones!

A·1 Quadratic Formula ⭐ Easy

DISCRIMINANT: b²−4ac tells you HOW MANY roots

📎 Mini Example Solve x² − 5x + 6 = 0 → (x−2)(x−3) = 0 → x = 2 or 3
Using formula: x = (5 ± √(25−24)) / 2 = (5±1)/2

⚠️ What is the value of the discriminant for the equation 2x² − 4x + 5 = 0 ?
Discriminant = b² − 4ac (a=2, b=−4, c=5)

Explanation: b² − 4ac = (−4)² − 4(2)(5) = 16 − 40 = −24.
⚠️ Trap: Students forget (−4)² = +16, not −16. Always square first!
Since discriminant < 0 → no real roots (two complex roots).

A·2 Exponent Rules ⭐ Easy

POWER RULE: (xᵃ)ᵇ = x^(a·b) — MULTIPLY the exponents!

📎 Mini Example (x³)⁴ = x¹² | x² · x⁵ = x⁷ (ADD when multiplying same base)

⚠️ Simplify: (3x²y³)²
Distribute the exponent to EVERY factor inside.

Explanation: (3x²y³)² = 3² · (x²)² · (y³)² = 9 · x⁴ · y⁶ = 9x⁴y⁶.
⚠️ Trap A: 3² = 9, NOT 3×2 = 6. Square the coefficient too!

A·3 Rational Exponents ⭐⭐ Medium

x^(m/n) = (ⁿ√x)ᵐ → denominator = ROOT, numerator = POWER

📎 Mini Example 8^2/3 = (³√8)² = 2² = 4

⚠️ Evaluate: 27^4/3

Explanation: 27^4/3 = (³√27)⁴ = 3⁴ = 81.
⚠️ Trap: Don't multiply 27×(4/3). First find cube root (3), then raise to the 4th power.

A·4 Logarithms ⭐⭐ Medium

log_b(x) = y ↔ b^y = x → LOG = "What POWER?"

📎 Mini Example log₂(8) = 3 because 2³ = 8

⚠️ Solve for x: log₃(x) = 4

Explanation: log₃(x) = 4 ↔ 3⁴ = x → x = 81.
⚠️ Trap D: log₄(64) = 3 is a different problem. Don't swap base and answer!

A·5 Complex Numbers ⭐⭐ Medium

i² = −1 → i¹=i, i²=−1, i³=−i, i⁴=1 (cycle of 4!)

📎 Mini Example (2+3i)(1−i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i

⚠️ Simplify: (3 + 2i)(3 − 2i)
Hint: This is a "difference of squares" pattern.

Explanation: (3+2i)(3−2i) = 3² − (2i)² = 9 − 4i² = 9 − 4(−1) = 9 + 4 = 13.
⚠️ Key: i² = −1, so −4i² = −4(−1) = +4. Result is always a real number!

A·6 Polynomial Factoring ⭐ Easy

SUM OF CUBES: a³+b³ = (a+b)(a²−ab+b²) — middle sign FLIPS!

📎 Mini Example x³ − 8 = (x−2)(x² + 2x + 4) [difference of cubes]

⚠️ Factor completely: x³ + 27

Explanation: a=x, b=3 → (x+3)(x²−3x+9).
Formula: a³+b³ = (a+b)(a²−ab+b²). The middle term is −ab = −3x. Sign is MINUS!
⚠️ Trap A: Middle sign should be NEGATIVE (−3x), not positive.

A·7 Functions — Inverse ⭐⭐ Medium

INVERSE: swap x and y, then solve for y → f⁻¹ reflects over y=x

📎 Mini Example f(x) = 2x+1 → swap: x = 2y+1 → y = (x−1)/2 → f⁻¹(x) = (x−1)/2

⚠️ Find f⁻¹(x) if f(x) = 3x − 6

Explanation: y = 3x−6 → swap: x = 3y−6 → x+6 = 3y → y = (x+6)/3.
⚠️ Trap D: +6 moves to the OTHER side as +6 (not −6). Add, don't subtract!

A·8 Arithmetic Sequences ⭐ Easy

aₙ = a₁ + (n−1)d → nth term = first + (n−1)×common difference

📎 Mini Example Seq: 3, 7, 11, ... → a₁=3, d=4 → a₁₀ = 3 + 9(4) = 39

⚠️ In the sequence 5, 8, 11, 14, … find the 20th term.

Explanation: a₁=5, d=3 → a₂₀ = 5 + (20−1)×3 = 5 + 57 = 62.
⚠️ Trap B: Using n=20 instead of (n−1)=19 gives 5+60=65. Remember n−1!

A·9 Systems of Equations ⭐⭐ Medium

ELIMINATION: multiply to make coefficients equal, then ADD or SUBTRACT

📎 Mini Example x+y=5 and x−y=1 → Add: 2x=6 → x=3 → y=2

⚠️ Solve the system: 2x + 3y = 12 and 4x − 3y = 6

Explanation: Add equations: 6x = 18 → x = 3.
Sub back: 2(3)+3y=12 → 6+3y=12 → y = 2. Answer: (3, 2).
⚠️ Always verify in BOTH equations: 4(3)−3(2) = 12−6 = 6 ✓

A·10 Geometric Series ⭐⭐ Medium

Sₙ = a₁(1−rⁿ)/(1−r) → r = common ratio (multiply each time)

📎 Mini Example Sum of 2+6+18+54 → a₁=2, r=3, n=4 → S₄ = 2(1−81)/(1−3) = 2(−80)/(−2) = 80

⚠️ Find the sum of the first 5 terms of 3 + 6 + 12 + 24 + ...

Explanation: a₁=3, r=2, n=5 → S₅ = 3(1−2⁵)/(1−2) = 3(1−32)/(−1) = 3(−31)/(−1) = 93.
⚠️ Check: 3+6+12+24+48 = 93 ✓ Trap B: forgetting to multiply by a₁.

📐 Geometry — 10 Core Questions

Diagrams are your best friend — sketch it out!

G·1 Triangle Angles ⭐ Easy

TRIANGLE: all angles sum to 180° — always!

📎 Mini Example Angles 50°, 70°, x° → x = 180 − 50 − 70 = 60°

⚠️ A triangle has angles 42° and 87°. What is the third angle?

Explanation: 42 + 87 + x = 180 → x = 180 − 129 = 51°.
⚠️ Trap: 42+87 = 129, NOT 139. Careful with addition!

G·2 Pythagorean Theorem ⭐ Easy

a² + b² = c² → c is ALWAYS the hypotenuse (longest side)

📎 Mini Example legs 3, 4 → 3²+4² = 9+16 = 25 → c = √25 = 5

⚠️ A right triangle has legs 5 and 12. Find the hypotenuse.

Explanation: 5²+12² = 25+144 = 169 → c = √169 = 13.
⚠️ 5-12-13 is a Pythagorean triple — memorize it! (Like 3-4-5)

G·3 Circle — Arc Length ⭐⭐ Medium

ARC = (θ/360°) × 2πr → fraction of the full circle

📎 Mini Example r=6, θ=90° → Arc = (90/360)×2π(6) = (1/4)×12π = 3π

⚠️ A circle has radius 10. Find the arc length for a central angle of 60°.
Leave answer in terms of π.

Explanation: Arc = (60/360)×2π(10) = (1/6)×20π = 20π/6 = 10π/3.
⚠️ Trap B: (1/6)×10 = 5π forgets the factor of 2 in circumference 2πr.

G·4 Similar Triangles ⭐⭐ Medium

SIMILAR: same angles, proportional sides → set up a proportion, cross-multiply

📎 Mini Example Triangles with sides 3:6 ratio → all sides scale by ×2

⚠️ △ABC ~ △DEF. AB = 8, BC = 12, DE = 6. Find EF.

Explanation: AB/DE = BC/EF → 8/6 = 12/EF → EF = (12×6)/8 = 72/8 = 9.
⚠️ Ratio: 8:6 = 4:3, so EF = 12×(3/4) = 9. Corresponding sides must match!

G·5 Volume — Cylinder ⭐ Easy

V = πr²h → "area of circle" × height

📎 Mini Example r=3, h=5 → V = π(9)(5) = 45π

⚠️ A cylinder has diameter 8 and height 7. Find the volume.
Careful: diameter ≠ radius!

Explanation: diameter=8 → r=4. V = π(4²)(7) = π(16)(7) = 112π.
⚠️ Trap A: Using d=8 as r gives π(64)(7)=448π. ALWAYS halve the diameter!

G·6 Parallel Lines & Transversal ⭐ Easy

ALTERNATE INTERIOR = equal | CO-INTERIOR (same-side) = 180°

📎 Mini Example Corresponding angles: both 55° (same position, same size) ✓

⚠️ Two parallel lines are cut by a transversal. One co-interior angle is 65°. Find its co-interior partner.
Co-interior = "same-side interior" = between the parallel lines, same side.

Explanation: Co-interior angles are supplementary → 65° + x = 180° → x = 115°.
⚠️ Trap A: 65° would be alternate interior (equal). Co-interior adds to 180°!

G·7 Special Right Triangles ⭐⭐ Medium

45-45-90: sides = x, x, x√2 | 30-60-90: sides = x, x√3, 2x

📎 Mini Example 30-60-90: short leg=5 → hyp=10, long leg=5√3

⚠️ In a 45°-45°-90° triangle, the hypotenuse is 8√2. Find a leg.

Explanation: In 45-45-90: hyp = leg × √2 → 8√2 = leg × √2 → leg = 8.
⚠️ Divide the hypotenuse by √2 to get the leg. Don't multiply!

G·8 Coordinate Geometry — Midpoint ⭐ Easy

MIDPOINT = average of x's, average of y's → M = ((x₁+x₂)/2, (y₁+y₂)/2)

📎 Mini Example A(2,4) B(8,10) → M = ((2+8)/2, (4+10)/2) = (5, 7)

⚠️ M is the midpoint of AB. A = (1, 3) and M = (4, 7). Find B.
Work backwards: M = (A+B)/2 → B = 2M − A

Explanation: B = 2M − A = (2×4−1, 2×7−3) = (8−1, 14−3) = (7, 11).
⚠️ Trap A: That's the midpoint formula forward. Here you need to reverse it!

G·9 Exterior Angle Theorem ⭐ Easy

EXTERIOR ANGLE = sum of TWO non-adjacent interior angles

📎 Mini Example Interior: 40° and 70° → Exterior = 40+70 = 110°

⚠️ An exterior angle of a triangle is 130°. One non-adjacent interior angle is 58°. Find the other non-adjacent angle.

Explanation: Exterior = sum of two remote interior → 130 = 58 + x → x = 72°.
⚠️ Trap B: 130−58=72, NOT 88. Double-check subtraction: 130−58 = 72 ✓

G·10 Circle — Inscribed Angle ⭐⭐ Medium

INSCRIBED ANGLE = HALF the intercepted arc (central angle = arc)

📎 Mini Example Arc = 80° → Inscribed angle = 40° (half the arc)

⚠️ An inscribed angle intercepts an arc of 148°. Find the inscribed angle.

Explanation: Inscribed angle = arc/2 = 148/2 = 74°.
⚠️ Trap A: That's the arc itself. Inscribed angle is ALWAYS half. Don't confuse with the central angle!

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