Math Practice Workbook

Math Workbook

Algebra 2 & Geometry · Self-Study Edition

✏️ For Independent Study

📐 Geometry 📊 Algebra 2 🧠 Memory Points ✅ Multiple Choice 💡 Worked Examples

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QUADRATIC FORMULA ★ Easy

⚡ Memory Point: discriminant = b²−4ac
If b²−4ac > 0 → 2 real roots | = 0 → 1 root | < 0 → no real roots

📝 EXAMPLE

Solve: $x^2 - 5x + 6 = 0$

→ $a=1,\ b=-5,\ c=6$ Discriminant: $(-5)^2 - 4(1)(6) = 1 > 0$

→ $x = \dfrac{5 \pm 1}{2}$ so $x = 3$ or $x = 2$ ✔

How many real solutions does $3x^2 - 2x + 5 = 0$ have?
(Hint: calculate the discriminant first!)

VERTEX FORM ★ Easy

⚡ Memory Point: Vertex form: y = a(x−h)² + k → vertex is (h, k)
Watch out: sign of h is FLIPPED inside the parentheses!

📝 EXAMPLE

$y = 2(x-3)^2 + 1$ → Vertex = (3, 1) not (−3, 1)!

What is the vertex of $y = -3(x + 4)^2 - 7$ ?

POLYNOMIAL FACTORING ★★ Medium

⚡ Memory Point: Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$
Sum of Cubes: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$

📝 EXAMPLE

Factor: $x^2 - 9 = (x+3)(x-3)$ ← both are perfect squares!

Which is the completely factored form of $2x^3 - 8x$ ?

RATIONAL EXPONENTS ★★ Medium

⚡ Memory Point: $x^{m/n} = \sqrt[n]{x^m}$ → denominator = index of root, numerator = power

📝 EXAMPLE

$8^{2/3} = (\sqrt[3]{8})^2 = (2)^2 = 4$

Tip: Root first, then power → easier numbers!

Evaluate: $27^{4/3}$

LOGARITHMS ★★ Medium

⚡ Memory Point: $\log_b x = y$ means $b^y = x$
"log asks: what POWER gives me x?"

📝 EXAMPLE

$\log_2 8 = ?$ → Ask: $2^? = 8$ → $2^3 = 8$ → Answer: 3

Which equation is equivalent to $\log_5 125 = 3$ ?

COMPLEX NUMBERS ★★★ Tricky

⚡ Memory Point: $i^2 = -1$ cycle: $i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1$ → repeats every 4!
To find $i^{big}$: divide exponent by 4, use remainder

📝 EXAMPLE

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

Simplify: $(2 + 3i)(2 - 3i)$

SYSTEMS OF EQUATIONS ★ Easy

⚡ Memory Point: Substitution: isolate one variable, plug into other equation
Elimination: add/subtract equations to cancel a variable

📝 EXAMPLE

$y = 2x + 1$ and $y = -x + 4$ → set equal: $2x+1 = -x+4$ → $x=1, y=3$

Solve the system: $y = x + 2$ and $y = -2x + 8$. What is $x$?

FUNCTION INVERSE ★★ Medium

⚡ Memory Point: To find inverse: swap x and y, then solve for y
Graph of inverse = reflection over y = x

📝 EXAMPLE

$f(x) = 2x + 3$ → swap: $x = 2y + 3$ → $y = \dfrac{x-3}{2}$

So $f^{-1}(x) = \dfrac{x-3}{2}$

Find $f^{-1}(x)$ if $f(x) = 5x - 10$

ARITHMETIC SEQUENCES ★ Easy

⚡ Memory Point: $a_n = a_1 + (n-1)d$ where d = common difference
"first term PLUS (steps × gap)"

📝 EXAMPLE

Sequence: 3, 7, 11, 15… $d = 4$

$a_{10} = 3 + (10-1)(4) = 3 + 36 = 39$ ✔

In the sequence 5, 8, 11, 14, …, what is the 20th term?

EXPONENTIAL GROWTH / DECAY ★★★ Tricky

⚡ Memory Point: Growth: $y = a(1+r)^t$ Decay: $y = a(1-r)^t$
If base > 1 → growth | base < 1 → decay

📝 EXAMPLE

Population doubles every year starting at 100:

$y = 100(2)^t$ → after 3 yrs: $100 \times 8 = 800$

A car worth $20,000 depreciates at 15% per year. Which formula models its value $V$ after $t$ years?

TRIANGLE ANGLE SUM ★ Easy

⚡ Memory Point: All triangle angles add to 180°
Exterior angle = sum of 2 non-adjacent interior angles

📝 EXAMPLE

Angles: 50°, 70°, ? → $50 + 70 + x = 180$ → $x = 60°$

A triangle has angles $2x + 10$°, $3x$°, and $x + 20$°. Find $x$.

PYTHAGOREAN THEOREM ★ Easy

⚡ Memory Point: $a^2 + b^2 = c^2$ (c = longest side = hypotenuse)
Common triples: 3-4-5, 5-12-13, 8-15-17

📝 EXAMPLE

Legs: 6 and 8 → $6^2 + 8^2 = 36 + 64 = 100$ → $c = 10$

A right triangle has legs of length $9$ and $12$. Find the hypotenuse.

CIRCLE: ARC & CENTRAL ANGLE ★★ Medium

⚡ Memory Point: Central angle° = arc degree (same number!)
Arc length = $\dfrac{\theta}{360} \times 2\pi r$

📝 EXAMPLE

Central angle = 90°, r = 8

Arc length = $\dfrac{90}{360} \times 2\pi(8) = \dfrac{1}{4} \times 16\pi = 4\pi$

A circle has radius 10. A central angle measures 72°. What is the arc length? (Leave in terms of $\pi$)

TRIANGLE CONGRUENCE (SSS / SAS / ASA) ★★ Medium

⚡ Memory Point: Valid: SSS, SAS, ASA, AAS, HL
NOT valid: SSA (the "donkey theorem" 🐴) and AAA (only shows similarity)

📝 EXAMPLE

Two triangles share: 2 sides and the included angle → SAS ✔

Two triangles have two equal angles and the non-included side between them equal. What congruence theorem applies?

PARALLEL LINES & TRANSVERSAL ★ Easy

⚡ Memory Point: Alternate interior = equal (Z-shape)
Co-interior (same-side) = supplementary → add to 180°
Corresponding = equal (F-shape)

📝 EXAMPLE

Parallel lines, transversal. One angle = 110°.

Co-interior angle = $180 - 110 = 70°$

Two parallel lines are cut by a transversal. One co-interior angle is 65°. What is the other co-interior angle?

SPECIAL RIGHT TRIANGLES ★★★ Tricky

⚡ Memory Point: 45-45-90: sides = $x : x : x\sqrt{2}$
30-60-90: sides = $x : x\sqrt{3} : 2x$
"Short leg × √3 = Long leg" for 30-60-90

📝 EXAMPLE

30-60-90 triangle, short leg = 5

Long leg = $5\sqrt{3}$, Hypotenuse = $10$

In a 45-45-90 triangle, the hypotenuse is $8\sqrt{2}$. What is the length of each leg?

VOLUME OF SOLIDS ★★ Medium

⚡ Memory Point: Pyramid & Cone = $\frac{1}{3}$ × (base area × height)
"One-third of the prism" = easy way to remember!

📝 EXAMPLE

Cone: r = 3, h = 4 → $V = \dfrac{1}{3}\pi(3)^2(4) = 12\pi$

A square pyramid has base side $6$ cm and height $9$ cm. What is its volume?

SIMILARITY & SCALE FACTOR ★★ Medium

⚡ Memory Point: If scale factor = $k$, then:
Perimeter ratio = k | Area ratio = k² | Volume ratio = k³

📝 EXAMPLE

Similar triangles, scale factor = 3

Area of big ÷ Area of small = $3^2 = 9$

Two similar rectangles have sides in ratio 3:5. What is the ratio of their areas?

INSCRIBED ANGLE THEOREM ★★★ Tricky

⚡ Memory Point: Inscribed angle = ½ × intercepted arc
Central angle = = intercepted arc
"Inscribed is always HALF of central on same arc"

📝 EXAMPLE

Arc = 100° → Central angle = 100° → Inscribed angle = 50°

A central angle intercepts an arc of 140°. An inscribed angle intercepts the same arc. What is the inscribed angle?

COORDINATE GEOMETRY: MIDPOINT & DISTANCE ★★ Medium

⚡ Memory Point: Midpoint: $\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$ — just AVERAGE the coordinates!
Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ — Pythagorean theorem in disguise!

📝 EXAMPLE

Points (1, 2) and (5, 8): Midpoint = $\left(3, 5\right)$

Distance = $\sqrt{(5-1)^2 + (8-2)^2} = \sqrt{16+36} = \sqrt{52}$

The midpoint of segment $\overline{AB}$ is $(3, 7)$. Point $A$ is $(1, 3)$. What are the coordinates of $B$?