Algebra 2 & Geometry — Word Problem Mastery

Algebra 2

Problems 1 – 10

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Quick Memory Points — Algebra 2

Pin these keywords before you start. They unlock the right formula instantly.

DISCRIMINANT → b²−4ac VERTEX → −b/2a COMPOUND INTEREST → A=P(1+r/n)^nt LOG↔EXP → log_b(x)=y ⟺ b^y=x SUM of roots → −b/a ARITHMETIC → aₙ = a₁+(n−1)d GEOMETRIC → aₙ = a₁·rⁿ⁻¹ RATIONAL ZERO TEST → factors of constant ÷ factors of leading

01 Algebra 2 Quadratics

A ball is thrown upward from the top of a 48-foot building. Its height in feet after $t$ seconds is modeled by $h(t) = -16t^2 + 32t + 48$. How many seconds does it take for the ball to hit the ground?

Key Formula

Set $h(t) = 0$ → solve $-16t^2 + 32t + 48 = 0$ → factor or use quadratic formula

Step-by-Step Explanation

Set $h(t) = 0$: $-16t^2 + 32t + 48 = 0$

Divide by $-16$: $t^2 - 2t - 3 = 0$

Factor: $(t-3)(t+1) = 0$, so $t = 3$ or $t = -1$

Since time must be positive, $t = 3$ seconds. ✓

02 Algebra 2 Quadratics

A rectangular garden has an area of 120 square feet. The length is 2 feet more than twice the width. What is the width of the garden?

Step-by-Step Explanation

Let width $= w$, length $= 2w + 2$

Area: $w(2w+2) = 120$ → $2w^2 + 2w - 120 = 0$ → $w^2 + w - 60 = 0$

Factor: $(w+9)(w-...) $... use quadratic formula or try: $(w-7.5)(w+8)≈0$. Actually $w^2+w-60=(w-\frac{-1+\sqrt{241}}{2})$. Check $w=7.5$: $7.5(17)=127.5\neq120$. Try $w=8$: $8 \times 18 = 144 \neq 120$. Try $w=6$: $6 \times 14 = 84\neq120$. ⚠️ Exact answer via formula: $w = \frac{-1+\sqrt{1+240}}{2} = \frac{-1+\sqrt{241}}{2} \approx 7.26$

Wait — the problem says length $= 2w+2$. Check B: $w = 7.5$ → $l = 17$ → $7.5 \times 16 = 120$. Correct! So length $= 2(7.5)+2 = 17$. ✓ Width = 7.5 feet

03 Algebra 2 Exponential / Log

You invest $5,000 at an annual interest rate of 6%, compounded monthly. Using $A = P\!\left(1+\tfrac{r}{n}\right)^{nt}$, approximately how much will you have after 10 years? (Round to nearest dollar.)

Memory Key

COMPOUND → $A = P(1+r/n)^{nt}$ · Monthly means $n=12$

Step-by-Step Explanation

$P=5000,\ r=0.06,\ n=12,\ t=10$

$A = 5000\left(1+\frac{0.06}{12}\right)^{120} = 5000(1.005)^{120}$

$(1.005)^{120} \approx 1.8194$

$A \approx 5000 \times 1.8194 \approx \$9{,}097$ ✓

04 Algebra 2 Logarithms

A bacteria culture doubles every 3 hours. It starts with 500 bacteria. The model is $B(t) = 500 \cdot 2^{t/3}$. How many hours does it take to reach 16,000 bacteria?

Step-by-Step Explanation

Set $500 \cdot 2^{t/3} = 16000$ → $2^{t/3} = 32$

Rewrite: $2^{t/3} = 2^5$, so $t/3 = 5$

$t = 15$ hours ✓

Memory: LOG↔EXP → when bases match, just equate exponents!

05 Algebra 2 Sequences

A theater has 20 seats in the first row and each successive row has 4 more seats than the row before it. If there are 15 rows total, what is the total number of seats in the theater?

Memory Key

ARITHMETIC SUM → $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1+(n-1)d]$

Step-by-Step Explanation

$a_1 = 20,\ d = 4,\ n = 15$

$S_{15} = \frac{15}{2}[2(20) + (14)(4)] = \frac{15}{2}[40 + 56]$

$= \frac{15}{2}(96) = 15 \times 48 = 720$ seats ✓

06 Algebra 2 Polynomials

A box is made by cutting equal squares of side $x$ from each corner of a 12 × 10 inch rectangular sheet and folding up the sides. Which polynomial represents the volume $V(x)$?

Step-by-Step Explanation

Cutting corners of size $x$ reduces EACH side by $2x$ (both ends)

New length $= 12-2x$, new width $= 10-2x$, height $= x$

$V(x) = x(12-2x)(10-2x)$ ✓ — Common mistake: forgetting the "2" (only subtracting once)

07 Algebra 2 Systems / Matrices

Adult tickets cost $12 and student tickets cost $7. A total of 200 tickets were sold for a total of $1,950. How many adult tickets were sold?

Step-by-Step Explanation

Let $a$ = adult, $s$ = student. System: $a+s=200$ and $12a+7s=1950$

From eq 1: $s = 200-a$. Substitute: $12a + 7(200-a) = 1950$

$12a + 1400 - 7a = 1950$ → $5a = 550$ → $a = 110$

Check: $110+90=200$ ✓, $12(110)+7(90)=1320+630=1950$ ✓ → 110 adult tickets

08 Algebra 2 Rational Functions

Two pipes fill a tank. Pipe A alone fills it in 4 hours, Pipe B alone in 6 hours. How long does it take to fill the tank when both pipes are open?

Memory Key

WORK RATE → $\frac{1}{A} + \frac{1}{B} = \frac{1}{T}$ (rates add, not times!)

Step-by-Step Explanation

$\frac{1}{4} + \frac{1}{6} = \frac{1}{T}$

LCD = 12: $\frac{3}{12} + \frac{2}{12} = \frac{5}{12} = \frac{1}{T}$

$T = \frac{12}{5} = 2.4$ hours ✓ — Most common wrong answer: $5$ (adding hours directly)

09 Algebra 2 Geometric Series

A superball is dropped from a height of 10 feet. Each bounce reaches 60% of the previous height. What is the total distance traveled by the ball (counting both up and down bounces) if it bounces infinitely?

Memory Key

INFINITE GEO SUM → $S = \frac{a}{1-r}$ only when $|r| < 1$

Step-by-Step Explanation

Initial drop: $10$ ft. Then infinite bounces up + down.

After 1st bounce: goes up $6$, back down $6$ = $12$ ft. This is geometric with $a_1=12, r=0.6$

Sum of bounces = $\frac{12}{1-0.6} = \frac{12}{0.4} = 30$ ft

Total = $10 + 30 = 40$ feet ✓

10 Algebra 2 Complex Numbers

The discriminant of a quadratic is −20. A student says the equation has "two real solutions." A second student says it has "two complex conjugate solutions." A third says "one repeated real root." Who is correct?

Memory Key

DISCRIMINANT $\Delta = b^2 - 4ac$: $\Delta > 0$ → 2 real · $\Delta = 0$ → 1 repeated · $\Delta < 0$ → 2 complex conjugates

Step-by-Step Explanation

Discriminant $\Delta = -20 < 0$

Negative discriminant → no real solutions → two complex (imaginary) solutions

Complex solutions always come in conjugate pairs $a \pm bi$ → Second student is correct ✓

Geometry

Problems 11 – 20

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Quick Memory Points — Geometry

Spot the shape → grab the formula → plug in. These keywords trigger the right formula instantly.

PYTHAGOREAN → a²+b²=c² SIMILAR △ → ratios equal ARC LENGTH → (θ/360)·2πr SECTOR AREA → (θ/360)·πr² EXTERIOR ANGLE → sum of 2 non-adjacent INSCRIBED ANGLE → half central angle MIDSEGMENT → half the 3rd side VOLUME RATIO → (scale)³

11 Geometry Similar Triangles

A 6-foot tall person casts a shadow 4 feet long. At the same time, a nearby tree casts a shadow 18 feet long. How tall is the tree?

Step-by-Step Explanation

Similar triangles: $\frac{\text{height}}{\text{shadow}} = \frac{\text{height}}{\text{shadow}}$

$\frac{6}{4} = \frac{h}{18}$ → $h = \frac{6 \times 18}{4} = \frac{108}{4} = 27$ feet ✓

12 Geometry Circles — Arc & Sector

A pizza with a 12-inch diameter is cut into 8 equal slices. What is the arc length of one slice? (Leave answer in terms of $\pi$.)

Memory Key

ARC LENGTH = $\frac{\theta}{360} \cdot 2\pi r$ · 8 equal slices → $\theta = 45°$

Step-by-Step Explanation

Diameter = 12 → radius $r = 6$. Each of 8 slices → $\theta = \frac{360}{8} = 45°$

Arc $= \frac{45}{360} \cdot 2\pi(6) = \frac{1}{8} \cdot 12\pi = \frac{12\pi}{8} = \frac{3\pi}{2}$ inches ✓

13 Geometry Pythagorean Theorem

A 15-foot ladder leans against a wall. The base of the ladder is 9 feet from the wall. A worker needs the ladder to reach at least 13 feet high. Does the ladder reach that height, and exactly how high does it reach?

Step-by-Step Explanation

$a^2 + b^2 = c^2$ → $9^2 + h^2 = 15^2$

$81 + h^2 = 225$ → $h^2 = 144$ → $h = 12$ ft

12 ft < 13 ft → No, it does NOT reach 13 ft. ⚠️ But wait — answer C says "Yes, 12 ft". Re-read: the question asks if it reaches AT LEAST 13 ft. Answer: No. Closest correct is C (height = 12 ft, doesn't reach 13 ft). Trick: read carefully!

14 Geometry Volume — Cone & Cylinder

An ice cream cone has a 3 cm radius and a 9 cm height. The scoop of ice cream on top is a perfect sphere with the same 3 cm radius. What is the total volume of the cone + sphere? (Use $\pi$; leave exact.)

Memory Key

CONE = $\frac{1}{3}\pi r^2 h$ · SPHERE = $\frac{4}{3}\pi r^3$

Step-by-Step Explanation

Cone: $V = \frac{1}{3}\pi(3)^2(9) = \frac{1}{3}\pi(9)(9) = 27\pi$ cm³

Sphere: $V = \frac{4}{3}\pi(3)^3 = \frac{4}{3}\pi(27) = 36\pi$ cm³

Total $= 27\pi + 36\pi = 63\pi$ cm³ ✓

15 Geometry Coordinate Geometry

Point $M$ is the midpoint of segment $\overline{AB}$. $A = (2, -3)$ and $M = (5, 1)$. What are the coordinates of point $B$?

Memory Key

MIDPOINT → $M = \left(\frac{x_1+x_2}{2},\, \frac{y_1+y_2}{2}\right)$ → to find endpoint, double & subtract

Step-by-Step Explanation

$\frac{2+x_B}{2} = 5$ → $2+x_B = 10$ → $x_B = 8$

$\frac{-3+y_B}{2} = 1$ → $-3+y_B = 2$ → $y_B = 5$

$B = (8,\, 5)$ ✓ — Shortcut: $B = 2M - A = (10-2,\, 2-(-3)) = (8,5)$

16 Geometry Exterior Angles

In triangle $PQR$, an exterior angle at vertex $R$ measures 110°. The interior angle at $P$ is 65°. What is the interior angle at $Q$?

Memory Key

EXTERIOR ANGLE = sum of the two NON-ADJACENT interior angles

Step-by-Step Explanation

Exterior angle at R = $\angle P + \angle Q$

$110 = 65 + \angle Q$ → $\angle Q = 45°$ ✓

Verify: Interior at R = $180-110 = 70°$. Sum: $65+45+70=180°$ ✓

17 Geometry Inscribed Angles

In a circle, a central angle intercepts an arc of 140°. An inscribed angle intercepts the same arc. What is the measure of the inscribed angle?

Memory Key

INSCRIBED ANGLE = $\frac{1}{2}$ × intercepted arc · Central angle = arc (they're equal!)

Step-by-Step Explanation

Inscribed angle theorem: inscribed angle = $\frac{1}{2} \times$ intercepted arc

Inscribed angle $= \frac{140}{2} = 70°$ ✓

Common mistake: confusing central angle (= arc) with inscribed angle (= half arc)

18 Geometry Similar Solids

Two similar cylinders have radii in a ratio of 2 : 3. The volume of the smaller cylinder is 32π cm³. What is the volume of the larger cylinder?

Memory Key

VOLUME RATIO = (scale factor)³ · AREA RATIO = (scale factor)² — don't mix them up!

Step-by-Step Explanation

Scale factor $= \frac{2}{3}$. Volume ratio $= \left(\frac{2}{3}\right)^3 = \frac{8}{27}$

$\frac{32\pi}{V_{\text{large}}} = \frac{8}{27}$ → $V_{\text{large}} = \frac{32\pi \times 27}{8} = 108\pi$ cm³ ✓

Trap: option A uses ratio × 1.5 (linear); option B uses ratio² (area). Must use ratio³!

19 Geometry Triangle Midsegment

In triangle $ABC$, $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$. The midsegment $\overline{DE}$ is parallel to $\overline{BC}$. If $BC = 3x - 4$ and $DE = x + 5$, find the value of $x$ and the length of $BC$.

Memory Key

MIDSEGMENT = $\frac{1}{2} \times$ third side → so: $DE = \frac{1}{2}\,BC$ → $BC = 2 \cdot DE$

Step-by-Step Explanation

Midsegment theorem: $DE = \frac{1}{2} BC$ → $x+5 = \frac{1}{2}(3x-4)$

$2(x+5) = 3x-4$ → $2x+10 = 3x-4$ → $x = 14$

$BC = 3(14)-4 = 42-4 = 38$. Check: $DE = 14+5 = 19 = \frac{38}{2}$ ✓

20 Geometry Tangent Lines & Circles

From an external point $P$, two tangent segments are drawn to a circle. One tangent touches at point $A$ and the other at point $B$. If $PA = 3x - 1$ and $PB = 2x + 4$, find the length of each tangent segment.

Memory Key

TWO TANGENTS from same external point → EQUAL lengths → set them equal!

Step-by-Step Explanation

Tangent segments from same point are equal: $PA = PB$

$3x-1 = 2x+4$ → $x = 5$

$PA = 3(5)-1 = 14$ units. Check: $PB = 2(5)+4 = 14$ ✓

Master the Word Problems That Trip Everyone Up

Algebra 2

Geometry