Math Practice · Algebra 2 & Geometry

Algebra 2

Quadratics, Logarithms & More

0 of 10 answered

Quadratic Equations

⚡

Memory Point

DISCRIMINANT RULE: b²−4ac > 0 → two real roots | = 0 → one real root | < 0 → no real roots

Example: For the equation $x^2 - 5x + 6 = 0$, the discriminant is $(-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0$, so it has two distinct real roots: $x = 2$ and $x = 3$.

A ball is thrown upward. Its height (in feet) after $t$ seconds is modeled by $h(t) = -16t^2 + 48t + 5$. How many seconds does it take for the ball to reach its maximum height?

📖 Step-by-Step Explanation

The maximum of a parabola $at^2 + bt + c$ occurs at the vertex: $t = -\dfrac{b}{2a}$.

Here $a = -16$, $b = 48$: $\quad t = -\dfrac{48}{2(-16)} = -\dfrac{48}{-32} = 1.5$ seconds.

Trap: Many students plug in $t = 3$ (when ball hits ground) instead of finding the vertex. The vertex gives the max height, not the landing time.

Systems of Equations

⚡

Memory Point

SUBSTITUTION: isolate one variable → plug into other equation → solve → back-substitute

Example: If $y = 2x + 1$ and $x + y = 7$, substitute: $x + (2x+1) = 7 \Rightarrow 3x = 6 \Rightarrow x = 2, y = 5$.

A school store sells pencils for $0.25 and pens for $0.75. On Monday, a student buys a total of 12 items and spends exactly $5.00. How many pens did the student buy?

📖 Step-by-Step Explanation

Let $p$ = pencils, $n$ = pens.
Equation 1 (items): $p + n = 12$
Equation 2 (money): $0.25p + 0.75n = 5.00$

From eq.1: $p = 12 - n$. Substitute: $0.25(12-n) + 0.75n = 5$
$3 - 0.25n + 0.75n = 5$ → $0.5n = 2$ → $n = 4$ pens.

Trap: Forgetting to multiply cents correctly, or mixing up which variable is which.

Exponential Growth & Decay

⚡

Memory Point

GROWTH: A = P(1 + r)ᵗ | DECAY: A = P(1 − r)ᵗ | r = rate as decimal

Example: $1,000 grows at 5% annually for 3 years: $A = 1000(1.05)^3 \approx \$1{,}157.63$.

A car was purchased for $24,000. It depreciates (loses value) at a rate of 15% per year. Which expression represents the car's value after 5 years?

📖 Step-by-Step Explanation

Depreciation = losing value, so use decay formula: $A = P(1 - r)^t$.
Rate $r = 0.15$, so the multiplier each year is $1 - 0.15 = 0.85$.
After 5 years: $A = 24000(0.85)^5 \approx \$10{,}649$.

Trap: Using $1.15$ instead of $0.85$ — that would mean the car gains value! Always ask: growing or shrinking?

Logarithms

⚡

Memory Point

LOG ↔ EXPO: log_b(x) = y means b^y = x | "log asks: what POWER?"

Example: $\log_2(8) = 3$ because $2^3 = 8$. The log just asks: "2 to what power equals 8?"

An earthquake measuring 6.0 on the Richter scale is 10 times stronger than one measuring 5.0. The formula is $M = \log\left(\dfrac{I}{I_0}\right)$. An earthquake has intensity $I = 100{,}000 \cdot I_0$. What is its Richter magnitude $M$?

📖 Step-by-Step Explanation

$M = \log\!\left(\dfrac{I}{I_0}\right) = \log(100{,}000) = \log(10^5) = 5$.

$\log_{10}(10^5) = 5$ because $10^5 = 100{,}000$.

Trap: Students often forget that $\log$ without a base means $\log_{10}$, and write $\log(100{,}000) = 6$ by counting digits instead of using powers of 10.

Rational Functions

⚡

Memory Point

WORK RATE: rate × time = 1 job | combined: 1/A + 1/B = 1/T

Example: If Pipe A fills a tank in 4 hrs and Pipe B in 6 hrs, together: $\frac{1}{4}+\frac{1}{6} = \frac{5}{12}$, so $T = \frac{12}{5} = 2.4$ hrs.

Machine A can complete a job in 6 hours. Machine B can complete the same job in 9 hours. If both machines work together, how long (in hours) will it take to complete the job?

📖 Step-by-Step Explanation

Combined rate: $\dfrac{1}{6} + \dfrac{1}{9} = \dfrac{3}{18} + \dfrac{2}{18} = \dfrac{5}{18}$ jobs per hour.

Time $T = \dfrac{1}{\frac{5}{18}} = \dfrac{18}{5} = 3.6$ hours.

Trap: Many students average: $\frac{6+9}{2} = 7.5$ hrs. WRONG! You must add the rates (1/time), not the times.

Arithmetic Sequences

⚡

Memory Point

nTH TERM: aₙ = a₁ + (n−1)d | SUM: Sₙ = n/2 × (a₁ + aₙ)

Example: Sequence: 3, 7, 11, … Here $a_1 = 3$, $d = 4$. The 10th term: $a_{10} = 3 + 9(4) = 39$.

A theater has seats arranged in rows. The first row has 20 seats, and each row has 3 more seats than the previous row. The theater has 15 rows. What is the total number of seats in the theater?

📖 Step-by-Step Explanation

$a_1 = 20$, $d = 3$, $n = 15$.
Last row: $a_{15} = 20 + 14(3) = 20 + 42 = 62$ seats.
Total: $S_{15} = \dfrac{15}{2}(20 + 62) = \dfrac{15}{2}(82) = 15 \times 41 = 615$ seats.

Trap: Using $n = 14$ instead of $n - 1 = 14$ in the nth-term formula. Remember: the common difference is added $(n-1)$ times, not $n$ times!

Geometric Sequences

⚡

Memory Point

GEOMETRIC nTH: aₙ = a₁ · r^(n−1) | multiply, don't add!

Example: 2, 6, 18, 54 … Here $r = 3$. The 5th term: $a_5 = 2 \cdot 3^4 = 2 \times 81 = 162$.

A bacteria colony starts with 500 bacteria and doubles every hour. After how many hours will the colony first exceed 10,000 bacteria?

📖 Step-by-Step Explanation

$a_n = 500 \cdot 2^{n-1}$. We need $500 \cdot 2^{n-1} > 10{,}000$
$2^{n-1} > 20$
Check: $2^4 = 16$ (not enough), $2^5 = 32 > 20$ ✓
So $n - 1 = 5$, meaning $n = 6$... wait: at hour 4, colony = $500 \times 2^4 = 8{,}000$. At hour 5: $500 \times 2^5 = 16{,}000 > 10{,}000$. ✓

Answer: 5 hours. Trap: forgetting that hour 0 = start, so doubling once = hour 1.

Complex Numbers

⚡

Memory Point

i² = −1 | i¹=i, i²=−1, i³=−i, i⁴=1 → cycle repeats every 4!

Example: $(2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 2 + i - 3(-1) = 5 + i$.

In an AC circuit problem, the impedance is calculated as $Z = (3 + 4i) + (1 - 2i)$. What is $|Z|$, the magnitude of the total impedance?

📖 Step-by-Step Explanation

First add: $(3+4i)+(1-2i) = 4 + 2i$.
Magnitude: $|Z| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$.

Trap: Adding the magnitudes separately $|3+4i| + |1-2i| = 5 + \sqrt{5}$. You must add the complex numbers FIRST, then take the magnitude.

Polynomial Functions

⚡

Memory Point

REMAINDER THEOREM: f(c) = remainder when f(x) ÷ (x−c) | just PLUG IN!

Example: Is $x = 2$ a root of $f(x) = x^3 - 3x + 2$? Check: $f(2) = 8 - 6 + 2 = 4 \neq 0$. No.

A rectangular box has a volume modeled by $V(x) = x^3 - 6x^2 + 11x - 6$. If one dimension is $(x - 1)$, which of the following is also a factor?

📖 Step-by-Step Explanation

Use the Factor Theorem: if $(x - c)$ is a factor, then $f(c) = 0$.
Test $x = 2$: $f(2) = 8 - 24 + 22 - 6 = 0$ ✓
Test $x = -2$: $f(-2) = -8 - 24 - 22 - 6 \neq 0$ ✗
Test $x = 4$: $f(4) = 64 - 96 + 44 - 6 = 6 \neq 0$ ✗

Full factorization: $(x-1)(x-2)(x-3)$. Trap: Students try long division without first testing with the Factor Theorem — much slower!

Inverse Functions

⚡

Memory Point

INVERSE: swap x and y, then solve for y → f⁻¹ "undoes" f(x)

Example: If $f(x) = 2x + 3$, find $f^{-1}(x)$: Swap → $x = 2y + 3$ → $y = \dfrac{x-3}{2}$. So $f^{-1}(x) = \dfrac{x-3}{2}$.

A Celsius-to-Fahrenheit formula is $F = \dfrac{9}{5}C + 32$. A weather report shows 77°F. What is this temperature in Celsius? (Use the inverse function.)

📖 Step-by-Step Explanation

Inverse: $C = \dfrac{5}{9}(F - 32)$.
Plug in $F = 77$: $C = \dfrac{5}{9}(77 - 32) = \dfrac{5}{9}(45) = \dfrac{225}{9} = 25°C$.

Trap: Subtracting 32 AFTER multiplying by $\frac{5}{9}$: $\frac{5}{9}(77) - 32 \approx 42.8 - 32 = 10.8°C$. Always subtract 32 FIRST when converting F → C!

Geometry

Triangles, Circles & Proofs

0 of 10 answered

Pythagorean Theorem

⚡

Memory Point

a² + b² = c² → c is ALWAYS the hypotenuse (longest side, opposite right angle)

Example: A 3-4-5 right triangle: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓

A ladder is leaning against a wall. The base of the ladder is 5 feet from the wall, and the ladder reaches 12 feet up the wall. What is the length of the ladder?

📖 Step-by-Step Explanation

The wall, ground, and ladder form a right triangle.
$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ feet.

This is the classic 5-12-13 Pythagorean triple! Trap: Some students add the legs directly: $5 + 12 = 17$. You must square, add, then square root!

Similar Triangles

⚡

Memory Point

SIMILAR △: corresponding sides are PROPORTIONAL → set up cross-multiply ratio

Example: Triangles with sides 3, 4, 5 and 6, 8, 10 are similar (ratio 1:2). Missing side: $\frac{3}{6} = \frac{x}{8} \Rightarrow x = 4$.

A 6-foot tall person casts a 4-foot shadow. At the same time, a nearby tree casts a 10-foot shadow. How tall is the tree?

📖 Step-by-Step Explanation

The sun creates similar triangles. Set up the proportion:
$\dfrac{\text{height}}{\text{shadow}} = \dfrac{6}{4} = \dfrac{h}{10}$
Cross multiply: $4h = 60$ → $h = 15$ feet.

Trap: Students sometimes flip the ratio: $\frac{4}{6} = \frac{h}{10}$, giving $h \approx 6.7$. Always keep height/shadow consistent on both sides!

Circle Area & Arc Length

⚡

Memory Point

ARC LENGTH = (θ/360) × 2πr | SECTOR AREA = (θ/360) × πr² | θ in degrees

Example: Circle radius 6, central angle 90°. Arc length = $\frac{90}{360} \times 2\pi(6) = \frac{1}{4} \times 12\pi = 3\pi$.

A sprinkler rotates 120° and waters a region with radius 9 feet. What is the area of the watered region? (Leave answer in terms of $\pi$.)

📖 Step-by-Step Explanation

Sector area $= \dfrac{120}{360} \times \pi r^2 = \dfrac{1}{3} \times \pi(9)^2 = \dfrac{81\pi}{3} = 27\pi$ ft².

Trap: Using the arc length formula instead of the area formula, or forgetting to square the radius. Area uses $r^2$; arc length uses $r$.

Volume of 3D Shapes

⚡

Memory Point

CONE = (1/3)πr²h | CYLINDER = πr²h | "cone is 1/3 of cylinder with same base and height"

Example: Cylinder: $r=3, h=5$ → $V = \pi(9)(5) = 45\pi$. Cone (same): $V = \frac{1}{3}(45\pi) = 15\pi$.

An ice cream cone has radius 3 cm and height 12 cm. A scoop of ice cream (sphere) with radius 3 cm sits on top. What is the total volume? ($V_{sphere} = \frac{4}{3}\pi r^3$)

📖 Step-by-Step Explanation

Cone: $V = \dfrac{1}{3}\pi(3)^2(12) = \dfrac{1}{3}\pi(9)(12) = 36\pi$ cm³
Sphere: $V = \dfrac{4}{3}\pi(3)^3 = \dfrac{4}{3}\pi(27) = 36\pi$ cm³
Total: $36\pi + 36\pi = 72\pi$ cm³

Trap: Forgetting the $\frac{1}{3}$ for cone, or using diameter instead of radius in the sphere formula.

Coordinate Geometry

⚡

Memory Point

MIDPOINT: M = ((x₁+x₂)/2, (y₁+y₂)/2) | DISTANCE: d = √[(Δx)²+(Δy)²]

Example: Midpoint of $(2, 4)$ and $(6, 8)$: $M = \left(\frac{8}{2}, \frac{12}{2}\right) = (4, 6)$.

On a coordinate map, a fire station is at $(1, 3)$ and a hospital is at $(7, 11)$. An ambulance must stop at the exact midpoint between them. What are the coordinates of the midpoint?

📖 Step-by-Step Explanation

$M = \left(\dfrac{1+7}{2},\, \dfrac{3+11}{2}\right) = \left(\dfrac{8}{2},\, \dfrac{14}{2}\right) = (4, 7)$

Trap: Adding without dividing: $(1+7, 3+11) = (8, 14)$. That's option D — double the actual midpoint! Always divide by 2.

Triangle Angle Sum & Exterior Angles

⚡

Memory Point

EXTERIOR ANGLE = sum of the TWO non-adjacent interior angles (remote interior angles)

Example: In a triangle with interior angles 50° and 70°, the exterior angle at the third vertex = $50° + 70° = 120°$.

In triangle $PQR$, $\angle P = 40°$ and $\angle Q = 65°$. A student extends side $QR$ past $R$ to create an exterior angle at $R$. What is the measure of that exterior angle?

📖 Step-by-Step Explanation

Interior angle at $R$: $180° - 40° - 65° = 75°$.
Exterior angle at $R$: $180° - 75° = 105°$.
Or use the shortcut: exterior angle $= 40° + 65° = 105°$ directly!

Trap: Students find the interior angle (75°) and forget to subtract from 180°. Or they compute $180 - 40 - 65$ but don't realize they found the interior, not exterior, angle.

Parallel Lines & Transversals

⚡

Memory Point

Z-shape = ALTERNATE (equal) | F-shape = CORRESPONDING (equal) | C-shape = CO-INTERIOR (add to 180°)

Example: Two parallel lines cut by a transversal: alternate interior angles are equal. If one is 55°, the other is also 55°.

Two parallel streets are cut by a diagonal road. One angle formed is $(3x + 15)°$ and its co-interior (same-side interior) angle is $(2x + 25)°$. Find $x$.

📖 Step-by-Step Explanation

Co-interior angles (same-side interior) are supplementary — they add to 180°.
$(3x + 15) + (2x + 25) = 180$
$5x + 40 = 180$
$5x = 140$
$x = 28$

Trap: Setting the angles equal (as if they were alternate interior angles): $3x+15 = 2x+25$ → $x=10$. Co-interior angles ADD to 180°, not equal each other!

Trigonometry — SOH CAH TOA

⚡

Memory Point

SOH: sin=Opp/Hyp | CAH: cos=Adj/Hyp | TOA: tan=Opp/Adj

Example: In a right triangle with hypotenuse 10 and angle 30°, the opposite side = $10 \times \sin(30°) = 10 \times 0.5 = 5$.

A ramp makes an angle of 20° with the ground. The ramp is 15 feet long. How high does the ramp rise off the ground? (Round to nearest tenth; $\sin 20° \approx 0.342$)

📖 Step-by-Step Explanation

The ramp length is the hypotenuse. The height is the opposite side to the 20° angle.
Use SOH: $\sin(20°) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{h}{15}$
$h = 15 \times 0.342 = 5.13 \approx 5.1$ feet.

Trap: Using cosine instead of sine — cosine gives the horizontal distance along the ground ($\approx 14.1$ ft), not the height.

Circles — Inscribed Angle Theorem

⚡

Memory Point

INSCRIBED ANGLE = ½ × (intercepted arc) | central angle = full arc

Example: If an inscribed angle intercepts an arc of 100°, the inscribed angle = $\frac{100°}{2} = 50°$.

In a circle, a central angle $\angle AOB = 140°$. An inscribed angle $\angle ACB$ intercepts the same arc $AB$, where $C$ is a point on the major arc. What is $\angle ACB$?

📖 Step-by-Step Explanation

The central angle equals the intercepted arc: arc $AB = 140°$.
The inscribed angle $= \dfrac{1}{2} \times 140° = 70°$.

Trap: Some students confuse minor and major arcs. If C is on the major arc, it intercepts the minor arc $AB = 140°$. If the central angle were the minor arc, answer would change. Always identify which arc is intercepted!

Surface Area

⚡

Memory Point

CYLINDER SA = 2πr² + 2πrh → "top + bottom + curved side" | slant height ≠ height for cone!

Example: Cylinder with $r=2, h=5$: $SA = 2\pi(4) + 2\pi(2)(5) = 8\pi + 20\pi = 28\pi$.

A water tank is a closed cylinder with radius 4 feet and height 10 feet. How many square feet of metal are needed to build it? (Leave answer in terms of $\pi$.)

📖 Step-by-Step Explanation

$SA = 2\pi r^2 + 2\pi r h = 2\pi(16) + 2\pi(4)(10) = 32\pi + 80\pi = 112\pi$ ft².

Trap: Forgetting the two circular caps ($2\pi r^2$) and only computing the lateral surface $2\pi rh = 80\pi$. The problem says "closed cylinder" — always include both ends!

Master the ProblemsThat Trick Everyone.

Quadratics, Logarithms & More

Triangles, Circles & Proofs

Master the Problems
That Trick Everyone.