Algebra 2
10 Problems
01
Quadratic Formula
⚡ MEMORY: "negative b, plus-minus root, over 2a" — sing it!
Worked Example: Solve \(x^2 - 5x + 6 = 0\) → \(x = \frac{5 \pm \sqrt{25-24}}{2} = \frac{5\pm1}{2}\) → \(x=3\) or \(x=2\)
📌 Tricky Trap
Students often forget to check the sign of the discriminant \(b^2 - 4ac\). If it's negative, roots are imaginary!
Solve: \(2x^2 - 7x + 3 = 0\)
📖 Step-by-Step Solution
Use \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) with \(a=2,\ b=-7,\ c=3\).Discriminant: \((-7)^2 - 4(2)(3) = 49 - 24 = 25\).
\(x = \frac{7 \pm 5}{4}\) → \(x = \frac{12}{4} = 3\) or \(x = \frac{2}{4} = \frac{1}{2}\).
02
Complex Numbers
⚡ MEMORY: \(i^2 = -1\) always. Treat \(i\) like a variable until you replace \(i^2\).
Example: \((2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5+i\)
Simplify: \((3 + 2i)(1 - 4i)\)
📖 Solution
\((3+2i)(1-4i) = 3 - 12i + 2i - 8i^2\)Replace \(i^2 = -1\): \(= 3 - 10i - 8(-1) = 3 + 8 - 10i = \mathbf{11 - 10i}\).
Common mistake: forgetting to replace \(i^2\) with \(-1\).
03
Logarithms
⚡ MEMORY: "log = exponent" → \(\log_b x = y\) means \(b^y = x\)
Example: \(\log_2 8 = 3\) because \(2^3 = 8\)
📌 Tricky Part
\(\log(AB) = \log A + \log B\) but \(\log(A+B) \neq \log A + \log B\). Don't split sums!
Evaluate: \(\log_3 81 - \log_3 9\)
📖 Solution
Use Quotient Rule: \(\log_3 81 - \log_3 9 = \log_3 \frac{81}{9} = \log_3 9\).Since \(3^2 = 9\), we get \(\log_3 9 = \mathbf{2}\).
04
Polynomial Factoring
⚡ MEMORY: Difference of cubes: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\) — "SOAP"
Same sign · Opposite sign · Always Positive
Factor completely: \(8x^3 - 27\)
📖 Solution
\(8x^3 - 27 = (2x)^3 - 3^3\). Apply \(a^3-b^3=(a-b)(a^2+ab+b^2)\):\(= (2x-3)((2x)^2 + (2x)(3) + 3^2) = \mathbf{(2x-3)(4x^2+6x+9)}\).
Check signs using SOAP: Same(-), Opposite(+), Always Positive(+).
05
Rational Expressions
⚡ MEMORY: Dividing fractions = multiply by the RECIPROCAL. "KCF: Keep · Change · Flip"
Simplify: \(\dfrac{x^2 - 4}{x^2 - x - 6} \div \dfrac{x+2}{x-3}\)
📖 Solution
Factor: \(\frac{(x-2)(x+2)}{(x-3)(x+2)} \times \frac{x-3}{x+2}\).Cancel \((x+2)\) and \((x-3)\): \(\frac{(x-2)\cancel{(x+2)}}{\cancel{(x-3)}\cancel{(x+2)}} \times \frac{\cancel{(x-3)}}{(x+2)}\).
Result: \(\frac{x-2}{x+2}\)... wait — re-check: \(\frac{(x-2)(x+2)}{(x+3)(x-2)} \cdot \frac{x-3}{x+2}\). Actually numerator factors: \(x^2-x-6=(x-3)(x+2)\). So result \(= \mathbf{1}\).
06
Exponential Growth
⚡ MEMORY: \(A = P(1 + r)^t\) — "Principal · Rate bumped · Time powered"
Example: $1000 at 5% for 2 years = \(1000(1.05)^2 = \$1102.50\)
A population of 500 bacteria doubles every 3 hours. How many bacteria after 9 hours?
📖 Solution
In 9 hours: \(\frac{9}{3} = 3\) doubling periods.\(A = 500 \times 2^3 = 500 \times 8 = \mathbf{4000}\) bacteria.
07
Systems of Equations
⚡ MEMORY: Elimination = "make one variable disappear by adding/subtracting"
Solve the system:
\(\begin{cases} 3x + 2y = 12 \\ x - y = 1 \end{cases}\)
📖 Solution
From eq 2: \(x = y + 1\). Substitute: \(3(y+1)+2y=12 \Rightarrow 5y=9 \Rightarrow y=\frac{9}{5}\).\(x = \frac{9}{5}+1 = \frac{14}{5}\). Answer: \(\left(\frac{14}{5},\, \frac{9}{5}\right)\).
08
Vertex Form
⚡ MEMORY: Vertex form \(y = a(x-h)^2 + k\) → vertex is \((h, k)\). Note: opposite sign!
What is the vertex of \(y = 2(x + 3)^2 - 5\)?
📖 Solution
\(y = 2(x-(-3))^2 + (-5)\). Compare to \(y=a(x-h)^2+k\).Here \(h = -3\), \(k = -5\). Vertex = \(\mathbf{(-3,\ -5)}\).
Trap: \(x+3\) means \(h = -3\), not \(+3\). The sign flips!
09
Arithmetic Sequences
⚡ MEMORY: \(a_n = a_1 + (n-1)d\) — "First term + (steps) × step size"
Example: Sequence 3, 7, 11, 15... → \(d=4\), so \(a_{10} = 3 + 9(4) = 39\)
The 5th term of an arithmetic sequence is 23, and the common difference is 4. What is the 1st term?
📖 Solution
\(a_5 = a_1 + (5-1)(4) = 23\)\(a_1 + 16 = 23\)
\(a_1 = \mathbf{7}\).
10
Rational Exponents
⚡ MEMORY: \(x^{m/n} = \sqrt[n]{x^m}\) — "denominator is the ROOT, numerator is the POWER"
📌 Tricky Trap
\(8^{2/3} \neq 8^2 \div 3\). First take the cube root: \(\sqrt[3]{8} = 2\), then square: \(2^2 = 4\). Always root first — it keeps numbers small!
Evaluate: \(27^{4/3}\)
📖 Solution
\(27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = \mathbf{81}\).Step 1: Cube root of 27 = 3. Step 2: Raise to the 4th power: \(3^4 = 81\).
Geometry
10 Problems
11
Pythagorean Theorem
⚡ MEMORY: \(a^2 + b^2 = c^2\) — "legs squared + legs squared = hypotenuse squared"
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17. Memorize these!
A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?
📖 Solution
\(7^2 + 24^2 = c^2\)\(49 + 576 = 625\)
\(c = \sqrt{625} = \mathbf{25}\).
This is the 7-24-25 Pythagorean triple!
12
Circle — Arc Length
⚡ MEMORY: Arc Length = \(\frac{\theta}{360°} \times 2\pi r\) — "fraction of the circle × circumference"
A circle has radius 10. What is the arc length of a 72° central angle? (Use \(\pi \approx 3.14\))
📖 Solution
\(\text{Arc} = \frac{72}{360} \times 2\pi(10) = \frac{1}{5} \times 20\pi = \mathbf{4\pi} \approx 12.57\).72° is exactly 1/5 of 360°, so arc = 1/5 of circumference.
13
Triangle Similarity
⚡ MEMORY: AA · SAS · SSS — "Two angles equal = always similar (AA rule)"
📌 Key Concept
Similar triangles have proportional sides. Set up ratios: \(\frac{\text{big}}{\text{small}} = \frac{\text{big}}{\text{small}}\)
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?
📖 Solution
Area ratio = (side ratio)².\(\frac{\text{Area}_\text{large}}{\text{Area}_\text{small}} = \left(\frac{5}{3}\right)^2 = \frac{25}{9}\).
\(\text{Area}_\text{large} = 27 \times \frac{25}{9} = 3 \times 25 = \mathbf{75 \text{ cm}^2}\).
Trap: Don't multiply by 5/3 — area scales by the SQUARE of the linear ratio!
14
Volume of Cone
⚡ MEMORY: Cone = \(\frac{1}{3} \times\) cylinder. "Cone gets one-third of the party"
A cone has radius 6 cm and height 10 cm. What is its volume? (Leave answer in terms of \(\pi\))
📖 Solution
\(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(6)^2(10) = \frac{1}{3}\pi(36)(10) = \frac{360\pi}{3} = \mathbf{120\pi \text{ cm}^3}\).
15
Coordinate Geometry
⚡ MEMORY: Midpoint = "average of x's, average of y's" — \(M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)\)
Point \(M(3, -1)\) is the midpoint of segment \(\overline{AB}\). If \(A = (7, 5)\), find \(B\).
📖 Solution
\(\frac{7 + x_B}{2} = 3 \Rightarrow x_B = 6 - 7 = -1\)\(\frac{5 + y_B}{2} = -1 \Rightarrow y_B = -2 - 5 = -7\)
\(B = \mathbf{(-1,\ -7)}\).
16
Parallel Lines & Transversals
⚡ MEMORY: Alternate interior angles = EQUAL. Co-interior (same-side) = SUPPLEMENTARY (add to 180°)
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \((3x + 10)°\) and the other is \((2x + 20)°\). Find \(x\).
📖 Solution
Co-interior angles are supplementary: they add to 180°.\((3x+10) + (2x+20) = 180\)
\(5x + 30 = 180 \Rightarrow 5x = 150 \Rightarrow x = \mathbf{30}\).
17
Special Right Triangles
⚡ MEMORY: 45-45-90: sides are \(x,\ x,\ x\sqrt{2}\) — "legs equal, hyp = leg × √2"
📌 Also know
30-60-90: sides are \(x,\ x\sqrt{3},\ 2x\) — "short leg, short×√3, double"
In a 45-45-90 triangle, the hypotenuse is \(8\sqrt{2}\). What is the length of each leg?
📖 Solution
In a 45-45-90 triangle: hypotenuse = leg × \(\sqrt{2}\).So: leg = \(\frac{\text{hypotenuse}}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = \mathbf{8}\).
18
Circle Theorems
⚡ MEMORY: Inscribed angle = HALF the central angle (same arc). "Inscribed gets half the credit"
An inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle?
📖 Solution
Inscribed Angle Theorem: inscribed angle = \(\frac{1}{2}\) × intercepted arc.\(\angle = \frac{140°}{2} = \mathbf{70°}\).
Trap: Many students confuse inscribed angle with central angle. Central angle = arc; inscribed angle = half arc.
19
Surface Area of Sphere
⚡ MEMORY: \(SA = 4\pi r^2\) — "Four circles wrapped around a ball"
A sphere has a diameter of 12 cm. What is its surface area? (Leave in terms of \(\pi\))
📖 Solution
Diameter = 12, so radius = 6.\(SA = 4\pi r^2 = 4\pi(6)^2 = 4\pi(36) = \mathbf{144\pi \text{ cm}^2}\).
Trap: Using diameter instead of radius! Always halve first.
20
Triangle Inequality
⚡ MEMORY: Any two sides must SUM to MORE than the third side. "The shortcut must be shorter!"
📌 Quick Test
Add the TWO SMALLEST sides. If their sum > biggest side, it's a valid triangle.
Which set of lengths CAN form a triangle?
📖 Solution
Check each: A: \(2+3=5 \not> 7\) ✗ B: \(1+5=6 \not> 6\) ✗ (must be strictly greater)C: \(5+7=12>9\) ✓ D: \(4+4=8 \not> 9\) ✗
Answer: \(\mathbf{5, 7, 9}\).