Units 1–5
Algebra 2 — Word Problems
Quadratics · Exponentials · Logarithms · Systems · Functions
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Memory Key
Quadratic Formula & Vertex
DISCRIMINANT VERTEX
b²−4ac > 0 → 2 real roots
b²−4ac = 0 → 1 root
b²−4ac < 0 → no real roots
vertex x = −b/(2a)
📘 Worked Example — Maximum Height
A ball's height: \(h(t)=-16t^2+64t+4\). Find the maximum height.
Vertex time: \(t=-\dfrac{64}{2(-16)}=2\ \text{s}\)
Max height: \(h(2)=-16(4)+64(2)+4=-64+128+4=\mathbf{68\ ft}\)
1 Easy
Rocket Launch
Topic: Quadratic Max / Vertex
A toy rocket's height in feet after \(t\) seconds:What is the maximum height the rocket reaches?
A \(86\ \text{ft}\)B \(96\ \text{ft}\)C \(106\ \text{ft}\)D \(116\ \text{ft}\)
📖 Explanation
Vertex at \(t=-\dfrac{80}{2(-16)}=\dfrac{80}{32}=2.5\ \text{s}\).
\(h(2.5)=-16(6.25)+80(2.5)+6=-100+200+6=\mathbf{106\ ft}\)
Always use \(t=-b/(2a)\) for max/min, then plug back in.
2 Easy
Fencing the Garden
Topic: Quadratic Max Area
A farmer has 120 ft of fencing for a rectangular garden against a barn (one side is free). Width \(w\), so \(A(w)=w(120-2w)\).What width maximizes area?
A \(20\ \text{ft}\)B \(40\ \text{ft}\)C \(30\ \text{ft}\)D \(60\ \text{ft}\)
📖 Explanation
\(A(w)=120w-2w^2\). Vertex: \(w=-\dfrac{120}{2(-2)}=30\ \text{ft}\).
Trap: Using all 4 sides! Only 2 widths use fencing; the barn covers the length.
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Memory Key
Exponential & Log Rules
GROWTH b>1 DECAY 0<b<1 LOG=EXPONENT
log_b(x)=y ↔ b^y=x
ln(e^x)=x
log(AB)=log A+log B
📘 Worked Example — Half-Life
Start: 800 g, half-life 4 years. Amount after 12 years?
\(A=800\cdot\!\left(\tfrac{1}{2}\right)^{12/4}=800\cdot\!\left(\tfrac{1}{2}\right)^3=800\cdot\tfrac{1}{8}=100\ \text{g}\)
12 years = 3 half-lives → 800 → 400 → 200 → 100 g
3 Easy
Radioactive Decay
Topic: Exponential Decay / Half-Life
A substance has a half-life of 5 years . You start with 320 grams .15 years ?
A \(160\ \text{g}\)B \(80\ \text{g}\)C \(40\ \text{g}\)D \(20\ \text{g}\)
📖 Explanation
15 years = 3 half-lives. \(320 \to 160 \to 80 \to \mathbf{40\ g}\)
\(A=320\cdot\!\left(\tfrac{1}{2}\right)^{15/5}=320\cdot\tfrac{1}{8}=40\ \text{g}\)
4 Easy
Investment Doubling
Topic: Compound Interest
You invest $500 at 6% compounded annually. \(A=500(1.06)^t\).After how many full years does the account first exceed $1,000 ?
A 10 yearsB 11 yearsC 12 yearsD 14 years
📖 Explanation
Need \((1.06)^t>2\). Take log: \(t>\dfrac{\ln 2}{\ln 1.06}\approx\dfrac{0.6931}{0.05827}\approx 11.9\)
First whole year = 12 . Check: \(500(1.06)^{12}\approx\$1006\). ✓
5 Easy
Logarithmic Equation
Topic: Solving Log Equations (Extraneous Solutions!)
Solve for \(x\):\[\log_2(x+3)+\log_2(x-1)=5\]Always check for extraneous solutions after solving!
A \(x=3\)B \(x=5\)C \(x=7\)D \(x=-7\) and \(x=5\)
📖 Explanation
Combine: \(\log_2[(x+3)(x-1)]=5\Rightarrow(x+3)(x-1)=32\)
\(x^2+2x-35=0\Rightarrow(x+7)(x-5)=0\Rightarrow x=-7\ \text{or}\ x=5\)
\(x=-7\) gives \(\log_2(-4)\) — undefined ❌. Only \(\mathbf{x=5}\) is valid.
EXTRANEOUS CHECK: Always plug solutions back into the original for log problems.
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Memory Key
Systems of Equations
SUBSTITUTION ELIMINATION
no solution → parallel lines
infinite → same line
mixture: (concentration)(volume) = amount
6 Easy
Mixing Solutions
Topic: Systems — Mixture Problems
A chemist mixes a 20% acid solution with a 50% acid solution to make 90 mL of 30% solution . How many mL of the 20% solution are used?
A 30 mLB 45 mLC 60 mLD 75 mL
📖 Explanation
System: \(x+y=90\) and \(0.2x+0.5y=27\). Substitute \(y=90-x\):
\(0.2x+0.5(90-x)=27\Rightarrow-0.3x=-18\Rightarrow x=60\)
7 Easy
Two Numbers
Topic: System with Quadratic
Two positive numbers have sum 10 and product 21 . What is the larger number?
A 5B 6C 7D 8
📖 Explanation
\(x+y=10,\ xy=21.\) Sub \(y=10-x\): \(x(10-x)=21\Rightarrow x^2-10x+21=0\)
\((x-3)(x-7)=0\Rightarrow x=3\ \text{or}\ 7.\) Larger = 7 .
8 Easy
Phone Plan Break-Even
Topic: Linear Inequalities
Plan A: $20/month + $0.10/text.Plan B: $35/month unlimited.Plan B cheaper?
A More than 100 textsB More than 150 textsC More than 200 textsD More than 350 texts
📖 Explanation
Plan B < Plan A: \(35<20+0.10t\Rightarrow15<0.10t\Rightarrow t>150\)
At exactly 150 texts both plans cost $35. Over 150, Plan B wins.
9 Medium
Not a Root
Topic: Rational Root Theorem / Factor Theorem
Given \(f(x)=x^3-7x+6\), which value is NOT a root?Tip: Try factors of 6: ±1, ±2, ±3, ±6
A \(x=1\)B \(x=2\)C \(x=-3\)D \(x=3\)
📖 Explanation
\(f(1)=1-7+6=0\)✓ \(f(2)=8-14+6=0\)✓ \(f(-3)=-27+21+6=0\)✓
\(f(3)=27-21+6=12\neq0\) ✗ → \(x=3\) is NOT a root .
Factored: \((x-1)(x-2)(x+3)\).
10 Medium
Composition Trap
Topic: Composite Functions — Order Matters!
Let \(f(x)=2x+1\) and \(g(x)=x^2-3\).Common mistake: computing \(g(f(3))\) instead!
A \(10\)B \(46\)C \(13\)D \(7\)
📖 Explanation
INSIDE OUT: Evaluate inner function first.
Step 1: \(g(3)=9-3=6\)
Step 2: \(f(6)=2(6)+1=\mathbf{13}\)
Trap answer: \(g(f(3))=g(7)=49-3=46\). Different order = different answer!
Units 1–6
Geometry — Word Problems
Triangles · Circles · Area & Volume · Coordinate Geometry
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Memory Key
Triangle Rules
PYTHAGOREAN SIMILAR EXTERIOR ANGLE
a²+b²=c² (right ▲ only)
angles sum = 180°
exterior = sum of 2 remote interiors
triples: 3-4-5 / 5-12-13 / 8-15-17
📘 Worked Example — Similar Triangles
A 6-ft person casts a 4-ft shadow. A tree casts a 22-ft shadow. Height of tree?
\(\dfrac{6}{4}=\dfrac{h}{22}\Rightarrow h=\dfrac{6\times22}{4}=33\ \text{ft}\)
1 Easy
Ladder Against a Wall
Topic: Pythagorean Theorem
A 13-ft ladder leans against a wall. The base is 5 ft from the wall . How high does the ladder reach?
A 8 ftB 12 ftC 10 ftD 11 ft
📖 Explanation
\(5^2+h^2=13^2\Rightarrow25+h^2=169\Rightarrow h^2=144\Rightarrow h=\mathbf{12\ ft}\)
5-12-13 is a Pythagorean triple — memorize it!
2 Easy
Exterior Angle
Topic: Triangle Angle Sum + Exterior Angle Theorem
In triangle \(ABC\): \(\angle A=47°\), \(\angle B=68°\). What is the exterior angle at \(C\) ?
A 65°B 115°C 180°D 245°
📖 Explanation
Interior \(C=180°-47°-68°=65°\). Exterior \(=180°-65°=\mathbf{115°}\).
SHORTCUT: Exterior angle = sum of 2 non-adjacent interiors = \(47°+68°=115°\). ✓
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Memory Key
Circle Formulas
CIRCUMFERENCE SECTOR INSCRIBED ANGLE
C = 2πr, A = πr²
arc = (θ/360)·2πr
sector area = (θ/360)·πr²
inscribed angle = ½ central angle
3 Easy
Pizza Slice Area
Topic: Sector Area
A circular pizza has diameter 16 inches , cut into 8 equal slices . What is the area of one slice? (Leave in terms of \(\pi\).)
A \(4\pi\ \text{in}^2\)B \(16\pi\ \text{in}^2\)C \(8\pi\ \text{in}^2\)D \(32\pi\ \text{in}^2\)
📖 Explanation
Diameter=16 → radius=8. Total: \(\pi(8)^2=64\pi\). One slice: \(\dfrac{64\pi}{8}=\mathbf{8\pi}\).
Trap: Using 16 as the radius. Always halve the diameter!
4 Easy
Pool Tile Border
Topic: Area — Composite Shapes
A rectangular pool is 20 ft × 12 ft . A tile border 2 ft wide surrounds it. What is the area of only the border ?
A 144 ft²B 160 ft²C 176 ft²D 240 ft²
📖 Explanation
Outer: \((20+4)(12+4)=24\times16=384\). Pool: \(20\times12=240\).
Border \(=384-240=\mathbf{176\ ft^2}\)
Key: 2 ft border adds 4 ft to EACH dimension (both sides).
5 Easy
Cylinder Volume
Topic: 3D Volume
A cylindrical tank: radius 3 m , height 10 m .\[V=\pi r^2 h\]Volume in terms of \(\pi\)?
A \(30\pi\ \text{m}^3\)B \(60\pi\ \text{m}^3\)C \(90\pi\ \text{m}^3\)D \(900\pi\ \text{m}^3\)
📖 Explanation
\(V=\pi(3)^2(10)=9\times10\times\pi=\mathbf{90\pi\ m^3}\)
Choice D trap: forgot to square radius. \(3^2=9\), not 3!
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Memory Key
Parallel Lines + Transversal
ALTERNATE INTERIOR = EQUAL
CO-INTERIOR = 180°
CORRESPONDING = EQUAL
Z-shape → alternate → equal
C-shape → co-interior → supplementary
6 Easy
Parallel Lines Angles
Topic: Co-interior Angles (Same-Side Interior)
Two parallel lines are cut by a transversal. One angle is \((3x+15)°\), its co-interior angle is \((2x+25)°\). Find \(x\).
A \(x=20\)B \(x=25\)C \(x=28\)D \(x=30\)
📖 Explanation
Co-interior angles are supplementary (sum = 180°):
\((3x+15)+(2x+25)=180\Rightarrow5x+40=180\Rightarrow x=\mathbf{28}\)
Trap: Setting them equal (that's for alternate interior). Co-interior = 180°!
7 Medium
Similar Triangles Perimeter
Topic: Similarity & Scale Factor
Triangle \(ABC\sim\) Triangle \(DEF\). Sides of \(ABC\): 6, 8, 10. Shortest side of \(DEF\): 9. Find the perimeter of \(DEF\) .
A 27B 30C 36D 40
📖 Explanation
Scale factor: \(\dfrac{9}{6}=\dfrac{3}{2}\). Perimeter \(ABC=24\).
Perimeter \(DEF=24\times\dfrac{3}{2}=\mathbf{36}\)
Rule: Perimeters scale by ratio. Areas scale by ratio². Volumes by ratio³.
8 Medium
Cone vs Cylinder
Topic: Volume Comparison
A cone and cylinder have the same radius and height . Cone volume = 60 cm³ . What is the cylinder volume ?\[V_\text{cone}=\tfrac{1}{3}\pi r^2h,\quad V_\text{cyl}=\pi r^2h\]
A 20 cm³B 120 cm³C 180 cm³D 240 cm³
📖 Explanation
Cylinder = 3 × cone (same base & height).
\(V_\text{cyl}=3\times60=\mathbf{180\ cm^3}\)
Trap B (120 = 2×): some students think it's double. It's triple .
9 Medium
Find Point B from Midpoint
Topic: Midpoint Formula (Reverse)
\(M\) is the midpoint of \(\overline{AB}\). \(A=(2,-4)\) and \(M=(5,1)\). Find \(B\).
A \((3.5,\ -1.5)\)B \((7,\ 6)\)C \((8,\ 6)\)D \((8,\ -6)\)
📖 Explanation
\(5=\dfrac{2+x_B}{2}\Rightarrow x_B=8\quad;\quad 1=\dfrac{-4+y_B}{2}\Rightarrow y_B=6\)
\(B=(8,6)\). MIDPOINT = AVERAGE. Reverse: multiply midpoint by 2, subtract A.
10 Medium
Inscribed Angle Theorem
Topic: Circle Angles — Most Commonly Missed!
A central angle intercepts an arc of 140° . An inscribed angle intercepts the same arc . Measure of the inscribed angle?
A 140°B 40°C 70°D 280°
📖 Explanation
Inscribed angle \(=\dfrac{140°}{2}=\mathbf{70°}\)
INSCRIBED = HALF CENTRAL. Central angle = arc. Inscribed angle = ½ arc.