Algebra 2
Word Problems
Q 01
Quadratic Equations
QUICK KEY: ax² + bx + c = 0 → x = (−b ± √(b²−4ac)) / 2a
Discriminant: b²−4ac > 0 → 2 real roots | = 0 → 1 root | < 0 → no real roots
A ball is thrown upward from the ground. Its height (in feet) after \(t\) seconds is given by
\[ h(t) = -16t^2 + 64t \]
After how many seconds does the ball hit the ground again?
✦ Explanation
Set \(h(t) = 0\): \(-16t^2 + 64t = 0\) → factor out \(-16t\): \(-16t(t - 4) = 0\).So \(t = 0\) (launch) or \(t = 4\) (lands). The ball hits the ground at \(t = 4\) seconds.
Q 02
Systems of Equations
SUBSTITUTION → isolate one variable, plug into the other equation
Always check your answer by substituting back into BOTH equations!
Two tickets cost a total of $28. The adult ticket costs $4 more than twice the child ticket. What is the price of the child ticket?
✦ Explanation
Let child = \(c\), adult = \(a\).Equation 1: \(a + c = 28\)
Equation 2: \(a = 2c + 4\)
Substitute: \((2c + 4) + c = 28\) → \(3c = 24\) → \(c = 8\).
The child ticket costs $8. (Check: adult = $20; $20 + $8 = $28 ✓)
Q 03
Exponential Functions
GROWTH: A(t) = A₀ · (1 + r)^t | DECAY: A(t) = A₀ · (1 − r)^t
r = rate as decimal | t = time periods
A town has a population of 5,000. It grows at a rate of 4% per year. What will the population be after 2 years? (Round to the nearest whole number.)
✦ Explanation
\(A = 5000 \cdot (1.04)^2 = 5000 \cdot 1.0816 = 5408\).Common mistake: adding 4% twice (= 5,400) ignores compound growth. The answer is 5,408.
Q 04
Rational Expressions
WORK RATE: 1/A + 1/B = 1/T · (A, B = individual times; T = together)
Find common denominator → solve for T
Pipe A fills a tank in 6 hours. Pipe B fills the same tank in 3 hours. How long does it take if both pipes work together?
✦ Explanation
\(\frac{1}{6} + \frac{1}{3} = \frac{1}{T}\) → \(\frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\) → \(T = 2\) hours.Together they fill the tank in 2 hours.
Q 05
Logarithms
log_b(x) = y ↔ b^y = x | "LOG UNDOES EXPONENT"
log(AB) = logA + logB | log(A/B) = logA − logB | log(Aⁿ) = n·logA
A population of bacteria doubles every hour. Starting from 100 bacteria, after how many hours will there be 3,200 bacteria?
Hint: solve \(100 \cdot 2^t = 3200\)
Hint: solve \(100 \cdot 2^t = 3200\)
✦ Explanation
\(100 \cdot 2^t = 3200\) → \(2^t = 32 = 2^5\) → \(t = 5\).After 5 hours: 100→200→400→800→1600→3200. ✓
Q 06
Quadratic — Vertex
VERTEX x-coordinate: x = −b / (2a) · plug back for y
MAX profit / height / area → always find the vertex!
A company's profit (in thousands of dollars) is modeled by
\[ P(x) = -2x^2 + 12x - 10 \]
where \(x\) is the number of units (in hundreds) sold. How many units (hundreds) maximize profit?
✦ Explanation
\(x = \dfrac{-12}{2(-2)} = \dfrac{-12}{-4} = 3\).Maximum profit occurs at \(x = 3\) (hundreds of units).
\(P(3) = -2(9) + 12(3) - 10 = -18 + 36 - 10 = 8\) thousand dollars.
Q 07
Arithmetic Sequences
nth TERM: aₙ = a₁ + (n−1)d | SUM: Sₙ = n(a₁ + aₙ)/2
d = common difference (add same number each time)
A theater has 20 seats in the first row. Each row has 3 more seats than the row in front. How many seats are in the 10th row?
✦ Explanation
\(a_1 = 20,\ d = 3,\ n = 10\).\(a_{10} = 20 + (10-1)(3) = 20 + 27 = 47\).
The 10th row has 47 seats.
Q 08
Geometric Sequences
nth TERM: aₙ = a₁ · rⁿ⁻¹ · r = common ratio (multiply same number)
Exponential ≈ geometric sequence — same idea, different form!
A bouncing ball reaches 48 inches on its first bounce. Each bounce reaches ½ of the previous height. What height does it reach on the 5th bounce?
✦ Explanation
\(a_1 = 48,\ r = \tfrac{1}{2},\ n = 5\).\(a_5 = 48 \cdot \left(\tfrac{1}{2}\right)^{4} = 48 \cdot \tfrac{1}{16} = 3\) inches.
Bounces: 48 → 24 → 12 → 6 → 3 inches ✓
Q 09
Radical Equations
RADICAL → isolate √ first, THEN square both sides
⚠ ALWAYS CHECK for extraneous solutions after squaring!
The speed \(v\) (in m/s) of a wave on a string is given by \(v = \sqrt{4T}\), where \(T\) is the tension (in Newtons). If the wave speed is 10 m/s, what is the tension?
✦ Explanation
\(\sqrt{4T} = 10\) → square both sides: \(4T = 100\) → \(T = 25\) N.Check: \(\sqrt{4 \times 25} = \sqrt{100} = 10\) ✓. Tension is 25 N.
Q 10
Absolute Value Inequalities
|x| < k → −k < x < k ("AND" → middle)
|x| > k → x < −k OR x > k ("OR" → outside) Think: distance from zero on a number line
|x| > k → x < −k OR x > k ("OR" → outside) Think: distance from zero on a number line
A machine is supposed to produce bolts 50 mm long. The acceptable error is at most 2 mm. Which inequality represents all acceptable lengths \(x\)?
✦ Explanation
"Distance of \(x\) from 50 is at most 2" → \(|x - 50| \leq 2\).This means \(48 \leq x \leq 52\). All bolts must be between 48 mm and 52 mm.
Geometry
Word Problems
Q 11
Pythagorean Theorem
a² + b² = c² · c = HYPOTENUSE (longest side, opposite right angle)
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
A ladder is leaning against a wall. The base is 6 feet from the wall, and the top touches the wall at a height of 8 feet. How long is the ladder?
✦ Explanation
\(c^2 = 6^2 + 8^2 = 36 + 64 = 100\) → \(c = 10\).This is the 6-8-10 triple (double of 3-4-5). The ladder is 10 feet long.
Q 12
Area of Composite Figures
COMPOSITE AREA → break into simple shapes, ADD or SUBTRACT
Rectangle: l×w | Triangle: ½bh | Circle: πr²
A rectangular garden is 10 m × 8 m. A semicircular fountain with diameter 4 m is built inside it. What is the remaining area of the garden? (Use \(\pi \approx 3.14\))
✦ Explanation
Rectangle area: \(10 \times 8 = 80\ \text{m}^2\).Semicircle: radius = 2 m → \(\frac{1}{2}\pi r^2 = \frac{1}{2}(3.14)(4) = 6.28\ \text{m}^2\).
Remaining: \(80 - 6.28 = \mathbf{73.72\ \text{m}^2}\).
Q 13
Similar Triangles
SIMILAR △ → corresponding sides PROPORTIONAL
Set up ratio: a/b = c/d → cross-multiply to solve
A tree casts a shadow of 15 feet. At the same time, a 5-foot person casts a shadow of 3 feet. How tall is the tree?
✦ Explanation
\(\dfrac{\text{tree}}{15} = \dfrac{5}{3}\) → tree \(= \dfrac{5 \times 15}{3} = 25\) feet.The tree is 25 feet tall.
Q 14
Circle — Arc Length
ARC LENGTH = (θ/360°) × 2πr · θ = central angle in degrees
Think: "fraction of the full circle's circumference"
A circular pizza has a radius of 12 inches. A slice is cut with a central angle of 60°. What is the arc length of the crust? (Leave answer in terms of \(\pi\))
✦ Explanation
\(\text{Arc} = \dfrac{60}{360} \times 2\pi(12) = \dfrac{1}{6} \times 24\pi = 4\pi\) inches.The crust arc length is \(4\pi\) inches ≈ 12.57 inches.
Q 15
Volume — Cylinder
V(cylinder) = πr²h · V(cone) = ⅓πr²h · V(sphere) = (4/3)πr³
Cone = ⅓ of cylinder with same base and height
A cylindrical water tank has a radius of 3 meters and a height of 5 meters. How many cubic meters of water can it hold? (Leave in terms of \(\pi\))
✦ Explanation
\(V = \pi r^2 h = \pi (3)^2 (5) = \pi \cdot 9 \cdot 5 = 45\pi\ \text{m}^3\).The tank holds \(45\pi \approx 141.4\) cubic meters.
Q 16
Angle Relationships — Parallel Lines
PARALLEL + TRANSVERSAL: Alternate interior = EQUAL · Co-interior = 180°
Corresponding angles: same position = EQUAL (like an "F" shape)
Two parallel streets are cut by a diagonal road (transversal). One angle formed is 65°. What is the measure of its co-interior (same-side interior) angle?
✦ Explanation
Co-interior (consecutive interior / same-side interior) angles are supplementary → they add to 180°.\(180° - 65° = 115°\).
Note: alternate interior angles would equal 65°, but co-interior angles sum to 180°.
Q 17
Special Right Triangles
45-45-90: sides = x : x : x√2 | 30-60-90: sides = x : x√3 : 2x
30-60-90 → short leg × 2 = hypotenuse; short leg × √3 = long leg
A ramp makes a 30° angle with the ground. The ramp is 10 feet long (hypotenuse). How high does the ramp rise? (vertical height)
✦ Explanation
30-60-90 triangle: hypotenuse = 10, so short leg = \(\frac{10}{2} = 5\) feet.The side opposite the 30° angle is the short leg = vertical height = 5 feet.
(The horizontal distance = \(5\sqrt{3}\) feet — a common mix-up!)
Q 18
Surface Area — Rectangular Prism
SA(box) = 2(lw + lh + wh) · "3 pairs of rectangles"
Visualize unfolding the box into a flat net — 6 faces total
A gift box measures 8 cm × 5 cm × 3 cm. How much wrapping paper (in cm²) is needed to cover it exactly?
✦ Explanation
\(SA = 2(lw + lh + wh) = 2(8 \times 5 + 8 \times 3 + 5 \times 3)\)\(= 2(40 + 24 + 15) = 2(79) = 158\ \text{cm}^2\).
Need 158 cm² of wrapping paper.
Q 19
Triangle Inequality & Congruence
EXTERIOR ANGLE = sum of the 2 NON-ADJACENT interior angles
Interior angles of a triangle ALWAYS sum to 180°
In a triangle, two interior angles measure 47° and 68°. A student draws an exterior angle at the third vertex. What is the measure of that exterior angle?
✦ Explanation
Third interior angle: \(180° - 47° - 68° = 65°\).Exterior angle = sum of the other two interior angles = \(47° + 68° = 115°\).
(Or: supplement of 65° = \(180° - 65° = 115°\).) Answer: 115°.
Q 20
Coordinate Geometry
DISTANCE: d = √[(x₂−x₁)² + (y₂−y₁)²] | MIDPOINT: ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂−y₁)/(x₂−x₁) · Parallel → same slope, Perpendicular → slopes multiply to −1
Two friends live at coordinates A(1, 2) and B(7, 10) on a map (each unit = 1 km). What is the straight-line distance between their homes?
✦ Explanation
\(d = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\ \text{km}\).The distance is 10 km.
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