Math Practice · Algebra 2 & Geometry

Q 01

Quadratic Equations · Projectile Motion

A ball is thrown upward from the ground. Its height (in feet) after $t$ seconds is given by

$h(t) = -16t^2 + 64t$

How many seconds does it take for the ball to return to the ground?

💡 KEY: "return to ground" → h(t) = 0 · Factor or use quadratic formula · Always pick the positive t value

📖 Explanation

Set $h(t) = 0$: $-16t^2 + 64t = 0$ Factor out $-16t$: $-16t(t - 4) = 0$ So $t = 0$ (launch) or $t = 4$ (lands). ✅ Answer: 4 seconds

Q 02

Systems of Equations · Mixture Problem

A chemist mixes a 20% acid solution with a 50% acid solution to make 90 mL of a 30% acid solution. How many mL of the 20% solution does she need?

💡 MIXTURE FORMULA: (concentration × volume) = acid amount · Let x = mL of 20% → other = (90 − x) mL of 50%

📖 Explanation

Let $x$ = mL of 20% solution. Then $(90-x)$ = mL of 50% solution. $0.20x + 0.50(90-x) = 0.30(90)$ $0.20x + 45 - 0.50x = 27$ $-0.30x = -18 \Rightarrow x = 60$ mL ✅

Q 03

Exponential Growth · Population

A town has a population of 5,000. It grows at 3% per year. Which expression gives the population after $t$ years?

💡 EXPONENTIAL GROWTH: P = P₀(1 + r)ᵗ · growth → (1 + r) · decay → (1 − r) · r must be decimal

📖 Explanation

Exponential growth formula: $P = P_0(1+r)^t$.
Here $P_0 = 5000$, $r = 0.03$ (3% as decimal). $P = 5000(1.03)^t$ ✅ A is linear (wrong model). C uses rate as base (wrong). D uses 30% (wrong rate).

Q 04

Logarithms · Inverse of Exponential

An investment doubles when $2 = (1.07)^t$. Which equation correctly solves for $t$?

💡 LOG TRICK: aˣ = b → x = log(b) / log(a) · Take log of BOTH sides · $\log(a^t) = t \cdot \log(a)$

📖 Explanation

Take $\log$ of both sides: $\log 2 = \log(1.07)^t = t \cdot \log 1.07$ Divide both sides by $\log 1.07$: $t = \dfrac{\log 2}{\log 1.07} \approx 10.24$ years ✅ A has the fraction flipped. C and D apply log incorrectly.

Q 05

Quadratic · Maximum Value (Vertex)

A farmer has 120 feet of fencing to enclose a rectangular garden along a river (no fence needed on the river side). What width $w$ maximizes the area?

$A = w(120 - 2w)$

💡 VERTEX x-coord = −b / 2a · Rewrite in standard form ax²+bx+c first · Max area at vertex

📖 Explanation

Expand: $A = 120w - 2w^2 = -2w^2 + 120w$. Here $a=-2$, $b=120$. $w = \dfrac{-b}{2a} = \dfrac{-120}{2(-2)} = \dfrac{-120}{-4} = 30$ ft ✅ Maximum area $= 30 \times (120-60) = 30 \times 60 = 1800$ sq ft.

Q 06

Rational Functions · Work Rate Problem

Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. How many hours does it take to fill the tank if both pipes work together?

💡 WORK RATE: 1/A + 1/B = 1/T · Each pipe's rate = 1/time · Add rates together

📖 Explanation

Combined rate: $\dfrac{1}{6} + \dfrac{1}{4} = \dfrac{2}{12} + \dfrac{3}{12} = \dfrac{5}{12}$ tank/hour. $T = \dfrac{1}{5/12} = \dfrac{12}{5} = 2.4$ hours ✅ Common trap: adding times (6+4=10) is WRONG. Add rates, not times.

Q 07

Arithmetic Sequences · Nth Term

A theater has 20 seats in the first row. Each row after has 3 more seats than the previous. How many seats are in the 15th row?

💡 aₙ = a₁ + (n−1)d · a₁ = first term · d = common difference · n = term number

📖 Explanation

$a_1 = 20$, $d = 3$, $n = 15$. $a_{15} = 20 + (15-1)(3) = 20 + 14 \times 3 = 20 + 42 = 62$ ✅ Trap: using $n=15$ instead of $(n-1)=14$ gives 65 (wrong!).

Q 08

Quadratic Formula · Discriminant

The discriminant of $ax^2 + bx + c = 0$ is $b^2 - 4ac$. For $2x^2 - 4x + 5 = 0$, what does the discriminant tell us about the roots?

💡 DISCRIMINANT: >0 → 2 real roots · =0 → 1 real root · <0 → no real roots (complex)

📖 Explanation

$a=2$, $b=-4$, $c=5$. $\Delta = (-4)^2 - 4(2)(5) = 16 - 40 = -24$ Since $\Delta < 0$, there are no real roots — two complex (imaginary) roots. ✅

Q 09

Geometric Sequences · Compound Interest

You deposit $\$1{,}000$ in an account earning 5% annual compound interest. What is the balance after 3 years? (Round to the nearest dollar.)

$A = P(1 + r)^n$

💡 COMPOUND INTEREST = P(1+r)ⁿ · P = principal · r = rate (decimal) · n = years · NOT P×r×n (that's simple interest!)

📖 Explanation

$P = 1000$, $r = 0.05$, $n = 3$. $A = 1000(1.05)^3 = 1000 \times 1.157625 \approx \$1{,}158$ ✅ $\$1150$ = simple interest (wrong model). $\$1200$ = using 20% (wrong rate).

Q 10

Radical Equations · Square Root

Solve for $x$:

\sqrt{3x + 1} = x - 1

Remember to check for extraneous solutions!

💡 RADICAL EQUATION: isolate √ → square both sides → CHECK back in original · Squaring can create fake solutions!

📖 Explanation

Square both sides: $3x+1 = (x-1)^2 = x^2-2x+1$ $x^2 - 5x = 0 \Rightarrow x(x-5) = 0 \Rightarrow x=0 \text{ or } x=5$ Check $x=0$: $\sqrt{1} = -1$? → $1 \ne -1$ ❌ Extraneous!
Check $x=5$: $\sqrt{16} = 4$? → $4 = 4$ ✅
Wait — re-examine: $3(5)+1=16$, $\sqrt{16}=4$, $5-1=4$ ✓. But also try $x=8$: $\sqrt{25}=4 \ne 7$. Let me redo: $x^2-5x=0$ gives $x=0$ or $x=5$. Only $x=5$ is valid.

Correction note: The correct answer is $x = 5$. Please re-select — the answer key above has been updated to reflect $x=8$ was a typo in this card. The verified answer to $\sqrt{3x+1}=x-1$ is $\boxed{x=5}$.

Q 11

Pythagorean Theorem · Right Triangle

A 13-foot ladder leans against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?

💡 PYTHAGOREAN THEOREM: a² + b² = c² · c = hypotenuse (longest side, opposite right angle) · Memorize: 3-4-5, 5-12-13, 8-15-17

📖 Explanation

Ladder = hypotenuse $c = 13$, base $a = 5$, height $b = ?$ $5^2 + b^2 = 13^2 \Rightarrow 25 + b^2 = 169 \Rightarrow b^2 = 144 \Rightarrow b = 12$ ft ✅ This is the classic 5-12-13 Pythagorean triple!

Q 12

Circle · Arc Length

A circle has a radius of 9 cm. A central angle measures 120°. What is the arc length? (Use $\pi \approx 3.14$)

\text{Arc Length} = \dfrac{\theta}{360°} \times 2\pi r

💡 ARC LENGTH: (angle/360) × circumference · Or use radians: s = rθ · 120° = one-third of full circle

📖 Explanation

$L = \dfrac{120}{360} \times 2\pi(9) = \dfrac{1}{3} \times 18\pi = 6\pi \approx 18.85$ cm ✅ 120° is exactly $\frac{1}{3}$ of 360°, so arc = $\frac{1}{3}$ of circumference = $\frac{1}{3}(18\pi) = 6\pi$.

Q 13

Similar Triangles · Proportions

Two similar triangles have sides in ratio 3 : 5. If the smaller triangle has an area of 27 cm², what is the area of the larger triangle?

💡 AREA RATIO = (side ratio)² · Side ratio 3:5 → Area ratio 9:25 · Volume ratio = (side ratio)³

📖 Explanation

Side ratio $= 3:5$, so area ratio $= 3^2 : 5^2 = 9 : 25$. $\dfrac{27}{A_{\text{large}}} = \dfrac{9}{25} \Rightarrow A_{\text{large}} = \dfrac{27 \times 25}{9} = 75$ cm² ✅ Common trap: multiplying by $\frac{5}{3}$ (using side ratio for area) gives 45 — WRONG!

Q 14

Coordinate Geometry · Midpoint & Distance

Points $A(2, 5)$ and $B(8, 13)$ are endpoints of a segment. What is the midpoint of $\overline{AB}$?

💡 MIDPOINT: M = ((x₁+x₂)/2, (y₁+y₂)/2) · Average the x's, average the y's · Simple!

📖 Explanation

$M = \left(\dfrac{2+8}{2},\ \dfrac{5+13}{2}\right) = \left(\dfrac{10}{2},\ \dfrac{18}{2}\right) = (5, 9)$ ✅ D is the sum (not divided by 2) — a very common mistake!

Q 15

Volume · Composite Solids

An ice cream cone has a hemisphere of ice cream on top. The cone has radius $r = 3$ cm and height $h = 8$ cm. What is the total volume?

V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h \quad V_{\text{hemisphere}} = \dfrac{2}{3}\pi r^3

💡 COMPOSITE VOLUME = add each part separately · Hemisphere = half of sphere = (2/3)πr³ · Same radius for both here!

📖 Explanation

Cone: $V_c = \frac{1}{3}\pi(3)^2(8) = \frac{1}{3}\pi(9)(8) = 24\pi$
Hemisphere: $V_h = \frac{2}{3}\pi(3)^3 = \frac{2}{3}\pi(27) = 18\pi$ $V_{\text{total}} = 24\pi + 18\pi = 42\pi$ cm³ ✅

Q 16

Angles · Parallel Lines & Transversal

Two parallel lines are cut by a transversal. One angle measures $(3x + 20)°$ and its co-interior (same-side interior) angle measures $(x + 40)°$. Find $x$.

💡 CO-INTERIOR (consecutive interior) angles are SUPPLEMENTARY → sum = 180° · Alternate interior = EQUAL · Corresponding = EQUAL

📖 Explanation

Co-interior angles are supplementary (add to 180°): $(3x+20) + (x+40) = 180$ $4x + 60 = 180 \Rightarrow 4x = 120 \Rightarrow x = 30$ ✅ Check: $3(30)+20 = 110°$ and $30+40 = 70°$; $110 + 70 = 180°$ ✓

Q 17

Special Right Triangles · 30-60-90

In a 30-60-90 triangle, the side opposite 30° has length 7. What is the length of the hypotenuse?

💡 30-60-90 ratio: x : x√3 : 2x · 30° side (shortest) = x · 60° side = x√3 · hypotenuse = 2x · DOUBLE the short side!

📖 Explanation

In a 30-60-90 triangle: sides are $x : x\sqrt{3} : 2x$.
The 30° side $= x = 7$. Hypotenuse $= 2x$. $\text{Hypotenuse} = 2(7) = 14$ ✅ $7\sqrt{3}$ is the 60° side. $7\sqrt{2}$ belongs to a 45-45-90 triangle (wrong type!).

Q 18

Circles · Inscribed Angle Theorem

A central angle $\angle AOB = 84°$ in a circle. What is the measure of the inscribed angle $\angle ACB$ that intercepts the same arc $\widehat{AB}$?

💡 INSCRIBED ANGLE = ½ × (intercepted arc) = ½ × central angle · Central angle is TWICE the inscribed angle on same arc

📖 Explanation

Inscribed Angle Theorem: inscribed angle $= \frac{1}{2} \times$ central angle (same arc). $\angle ACB = \dfrac{1}{2} \times 84° = 42°$ ✅ A common trap is confusing inscribed and central angles and writing 84° or doubling to 168°.

Q 19

Surface Area · Cylinder

A cylindrical can has radius $r = 4$ cm and height $h = 10$ cm. What is the total surface area?

SA = 2\pi r^2 + 2\pi rh

💡 CYLINDER SA = 2 circles + lateral face = 2πr² + 2πrh · Factor: 2πr(r + h) · Don't forget BOTH circular bases!

📖 Explanation

$r=4$, $h=10$. $SA = 2\pi(4)^2 + 2\pi(4)(10) = 2\pi(16) + 2\pi(40) = 32\pi + 80\pi = 112\pi$ cm² ✅ $80\pi$ = lateral area only (forgot bases). $96\pi$ = added only one base.

Q 20

Trigonometry · SOHCAHTOA

From the top of a cliff 50 m high, the angle of depression to a boat is 30°. How far is the boat from the base of the cliff (horizontal distance)?

💡 SOHCAHTOA: sin=opp/hyp · cos=adj/hyp · tan=opp/adj · Angle of depression = angle of elevation (alternate interior angles) · tan(θ) = opposite/adjacent

📖 Explanation

The angle of depression from the cliff = angle of elevation from the boat = 30°.
Opposite = height = 50 m. Adjacent = horizontal distance $d$. $\tan 30° = \dfrac{50}{d} \Rightarrow d = \dfrac{50}{\tan 30°} = \dfrac{50}{1/\sqrt{3}} = 50\sqrt{3}$ m ✅ $\tan 30° = \frac{1}{\sqrt{3}} \approx 0.577$, so $d \approx 86.6$ m.

Algebra 2 & Geometry

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