Algebra 2 & Geometry — Study Guide

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Algebra 2

Quadratics · Polynomials · Logarithms · Systems · Sequences

Quadratic Equations

⬤ Easy — High Frequency

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Quick memory anchor ▸

DISCRIMINANT: b²−4ac → (+) two real, (0) one real, (−) no real

How many real solutions does the equation 2x² − 4x + 5 = 0 have?

📌 Example to understand first

For x² − 5x + 6 = 0, the discriminant is b²−4ac = 25−24 = 1 > 0 → two real solutions: x = 2 and x = 3.

Now apply the same method to 2x² − 4x + 5 = 0.

Quadratic Formula

⬤ Easy — Confusing Signs

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Quick memory anchor ▸

QUAD FORMULA: x = (−b ± √(b²−4ac)) / 2a

Solve x² + 6x + 5 = 0 using the quadratic formula. What are the solutions?

📌 Step-by-step model

For x² + 4x + 3 = 0: a=1, b=4, c=3
x = (−4 ± √(16−12)) / 2 = (−4 ± 2) / 2
→ x = −1 or x = −3

Vertex Form of a Parabola

⬤ Easy — Very Tricky Sign

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Quick memory anchor ▸

VERTEX FORM: y = a(x−h)²+k → vertex is (h, k), NOT (−h, k)!

The function f(x) = 3(x − 4)² + 7 is in vertex form. What is the vertex?

📌 Common mistake example

f(x) = 2(x + 3)² − 1 → vertex is (−3, −1), NOT (+3, −1).
Because (x + 3) = (x − (−3)), so h = −3.

Polynomial Long Division / Factor Theorem

⬤ Medium — Often Missed

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Quick memory anchor ▸

FACTOR THEOREM: f(a)=0 ↔ (x−a) is a factor

If f(x) = x³ − 2x² − 5x + 6, which of the following is a factor of f(x)?

📌 How to test a factor

Test x = 1: plug into f(1) = 1 − 2 − 5 + 6 = 0 → yes! So (x−1) is a factor.
Test x = −2: f(−2) = −8 − 8 + 10 + 6 = 0 → also a factor!

Logarithm Laws

⬤ Easy — Confusing Laws

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Quick memory anchor ▸

LOG PRODUCT: log(AB) = log A + log B · POWER: log(Aⁿ) = n·log A

Simplify: log₂(8) + log₂(4)

📌 Worked example

log₂(4) + log₂(8) = log₂(4 × 8) = log₂(32) = 5
Because 2⁵ = 32.

Exponential Equations

⬤ Easy — Same Base Trick

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Quick memory anchor ▸

SAME BASE: aˣ = aʸ → x = y (just match exponents)

Solve for x: 3^(2x−1) = 27

📌 Key insight

Rewrite 27 = 3³, so 3^(2x−1) = 3³.
Then 2x − 1 = 3 → x = 2.

Arithmetic Sequences

⬤ Easy — Off-by-one Error

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Quick memory anchor ▸

nth TERM: aₙ = a₁ + (n−1)d · careful: it's (n−1), not n!

An arithmetic sequence has a₁ = 5 and common difference d = 3. What is the 10th term?

📌 Common error alert

Wrong: a₁₀ = 5 + 10 × 3 = 35 ← uses n, not n−1.
Correct: a₁₀ = 5 + (10−1) × 3 = 5 + 27 = 32.

Geometric Sequences

⬤ Easy — Ratio Confusion

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Quick memory anchor ▸

GEO nth TERM: aₙ = a₁ · r^(n−1) · find r = a₂/a₁

A geometric sequence starts: 2, 6, 18, 54, …. What is the 6th term?

📌 Step by step

r = 6/2 = 3, a₁ = 2
a₆ = 2 × 3^(6−1) = 2 × 3⁵ = 2 × 243 = 486

Systems of Equations

⬤ Easy — Substitution Slip

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Quick memory anchor ▸

SUBSTITUTION: isolate one variable → plug in → solve → back-substitute

Solve the system: $y = 2x - 1 3x + y = 14$ What is the value of x?

Rational Exponents

⬤ Medium — Fraction Power Rule

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Quick memory anchor ▸

RATIONAL EXP: x^(m/n) = (ⁿ√x)ᵐ · denominator = root, numerator = power

Simplify: 27^(2/3)

📌 Worked model

8^(2/3) = (³√8)² = 2² = 4
Step 1: cube root first → ³√8 = 2
Step 2: then square → 2² = 4

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Geometry

Triangles · Circles · Coordinate Geometry · Area · Proofs

Pythagorean Theorem

⬤ Easy — Which Side is the Hypotenuse?

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Quick memory anchor ▸

PYTHAGORAS: a²+b²=c² · c is ALWAYS the hypotenuse (longest side)

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

📌 Pattern to memorize

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.
If you see 6 and 8 → multiply 3-4-5 by 2 → 6-8-10.

Circle — Area & Circumference

⬤ Easy — Mixing Up Formulas

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Quick memory anchor ▸

CIRCLES: Area = πr² · Circumference = 2πr · r = radius (not diameter!)

A circle has a diameter of 10 cm. What is its area? (Leave answer in terms of π)

📌 Trap: diameter vs. radius

Diameter = 10 → radius r = 5.
Area = π × 5² = 25π. Never use diameter directly in πr²!

Triangle — Interior Angle Sum

⬤ Easy — Algebra Inside Geometry

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Quick memory anchor ▸

TRIANGLE SUM: all 3 angles ALWAYS add to 180°

In a triangle, two angles measure 52° and 3x°. The third angle is 2x°. Find the value of x.

Special Right Triangles (30-60-90)

⬤ Medium — Ratio Mix-up

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Quick memory anchor ▸

30-60-90: sides = x : x√3 : 2x · short leg × 2 = hypotenuse

In a 30-60-90 triangle, the hypotenuse is 14. What is the length of the shorter leg?

📌 Side ratio chart

Short leg = x, Long leg = x√3, Hypotenuse = 2x.
So: 2x = 14 → x = 7.

Similar Triangles

⬤ Easy — Setting Up Proportion

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Quick memory anchor ▸

SIMILAR △: corresponding sides are PROPORTIONAL · match same position sides

Two similar triangles have corresponding sides in a ratio of 3 : 5. If the shorter triangle has a side of length 9, what is the corresponding side in the larger triangle?

Arc Length & Sector Area

⬤ Medium — Degrees vs. Radians

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Quick memory anchor ▸

ARC LENGTH: (θ/360) × 2πr · SECTOR AREA: (θ/360) × πr²

A circle has radius 9. A sector has a central angle of 120°. What is the area of the sector? (Leave in terms of π)

📌 Formula check

Sector area = (120/360) × π × 9²
= (1/3) × 81π = 27π

Coordinate Geometry — Midpoint

⬤ Easy — Average the Coordinates

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Quick memory anchor ▸

MIDPOINT: M = ((x₁+x₂)/2 , (y₁+y₂)/2) · just average x, then average y

Point A is at (2, −4) and point B is at (8, 6). What is the midpoint of segment AB?

Volume of a Cylinder

⬤ Easy — Radius vs. Diameter Trap

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Quick memory anchor ▸

CYLINDER VOLUME: V = πr²h · always halve the diameter to get r first

A cylinder has a diameter of 6 cm and a height of 10 cm. What is its volume? (Leave in terms of π)

Parallel Lines Cut by Transversal

⬤ Easy — Angle Pair Names

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Quick memory anchor ▸

PARALLEL LINES: Alternate interior = EQUAL · Co-interior (same-side) = 180°

Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures 115°. What is the measure of the other co-interior angle?

📌 Which pair equals 180°?

Co-interior angles (same side of the transversal, between the parallel lines) are supplementary: they add to 180°.
Alternate interior angles (opposite sides) are equal.

Distance Formula

⬤ Easy — Pythagoras in Disguise

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Quick memory anchor ▸

DISTANCE: d = √((x₂−x₁)²+(y₂−y₁)²) · it's just Pythagoras on a grid

What is the distance between points P(1, 2) and Q(4, 6)?

📌 Step by step

Δx = 4−1 = 3, Δy = 6−2 = 4
d = √(3² + 4²) = √(9+16) = √25 = 5