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Algebra 2
Quadratics · Systems · Logarithms · Complex Numbers · Sequences
Q 01
Example: Solve \(x^2 - 5x + 6 = 0\).
Here \(a=1, b=-5, c=6\). Discriminant: \((-5)^2 - 4(1)(6) = 25 - 24 = 1\).
So \(x = \dfrac{5 \pm 1}{2}\), giving \(x=3\) or \(x=2\). ✓
Now solve: What are the solutions to \(2x^2 + 3x - 2 = 0\)?
Here \(a=1, b=-5, c=6\). Discriminant: \((-5)^2 - 4(1)(6) = 25 - 24 = 1\).
So \(x = \dfrac{5 \pm 1}{2}\), giving \(x=3\) or \(x=2\). ✓
Now solve: What are the solutions to \(2x^2 + 3x - 2 = 0\)?
📖 Explanation
With \(a=2, b=3, c=-2\):\(x = \dfrac{-3 \pm \sqrt{9 + 16}}{4} = \dfrac{-3 \pm 5}{4}\)
So \(x = \dfrac{2}{4} = \dfrac{1}{2}\) or \(x = \dfrac{-8}{4} = -2\). ✅ Answer: A
Q 02
Watch out! Many students confuse "no real roots" with "no roots at all." Complex roots are still roots — they're just not on the number line!
How many real solutions does \(3x^2 - 4x + 5 = 0\) have?
How many real solutions does \(3x^2 - 4x + 5 = 0\) have?
Quick Check Strategy
Always calculate the discriminant FIRST before solving. \(\Delta = b^2 - 4ac\). If \(\Delta < 0\), stop — you know the answer immediately.
📖 Explanation
\(\Delta = (-4)^2 - 4(3)(5) = 16 - 60 = -44\)Since \(\Delta < 0\), there are no real solutions. The two solutions are complex conjugates. ✅ Answer: C
Q 03
Common mistake: Students write \(\log(A + B) = \log A + \log B\). This is WRONG! The product rule only applies to multiplication inside the log.
Simplify: \(\log_2 48 - \log_2 3\)
Simplify: \(\log_2 48 - \log_2 3\)
📖 Explanation
Using the quotient rule: \(\log_2 48 - \log_2 3 = \log_2 \dfrac{48}{3} = \log_2 16\)Since \(2^4 = 16\), the answer is \(\log_2 16 = 4\). ✅ Answer: A
Q 04
Tricky! Multiplying complex numbers uses FOIL just like polynomials — but don't forget to replace \(i^2\) with \(-1\) at the end.
Compute \((3 + 2i)(1 - 4i)\).
Compute \((3 + 2i)(1 - 4i)\).
FOIL Reminder
\((a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac-bd) + (ad+bc)i\)
📖 Explanation
\((3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i)\)\(= 3 - 12i + 2i - 8i^2\)
\(= 3 - 10i - 8(-1)\)
\(= 3 - 10i + 8 = 11 - 10i\) ✅ Answer: A
Q 05
Watch out: Always plug your answer BACK IN to check both equations. One equation satisfied ≠ solution!
Solve the system: \(\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}\)
Solve the system: \(\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}\)
📖 Explanation
Substitute \(y = 2x+1\) into \(3x + y = 16\):\(3x + (2x+1) = 16 \Rightarrow 5x = 15 \Rightarrow x = 3\)
Then \(y = 2(3)+1 = 7\). Check: \(3(3)+7 = 16\) ✓ ✅ Answer: A
Q 06
Confusing part: It's \((n-1)d\), not \(nd\)! The first term already exists, so you only add the common difference \((n-1)\) more times.
Find the 20th term of the arithmetic sequence: \(5, 9, 13, 17, \ldots\)
Find the 20th term of the arithmetic sequence: \(5, 9, 13, 17, \ldots\)
📖 Explanation
Here \(a_1 = 5\), \(d = 4\).\(a_{20} = 5 + (20-1)(4) = 5 + 19 \times 4 = 5 + 76 = 81\) ✅ Answer: B
Q 07
Trick alert: Students often mix up the formula with \(r^n\) vs \(r^{n-1}\). The sum formula uses \(r^n\) (not \(n-1\)!).
Find the sum of the first 5 terms of the geometric sequence: \(2, 6, 18, 54, \ldots\)
Find the sum of the first 5 terms of the geometric sequence: \(2, 6, 18, 54, \ldots\)
📖 Explanation
\(a_1 = 2,\ r = 3,\ n = 5\)\(S_5 = \dfrac{2(1 - 3^5)}{1 - 3} = \dfrac{2(1-243)}{-2} = \dfrac{2(-242)}{-2} = 242\) ✅ Answer: A
Q 08
Most common mistake: For \(y = (x-3)^2 + 2\), students say the vertex is \((-3, 2)\). It's actually \((3, 2)\)! The \(h\) value has a hidden sign flip.
What is the vertex of \(y = -2(x + 4)^2 - 1\)?
What is the vertex of \(y = -2(x + 4)^2 - 1\)?
📖 Explanation
Rewrite: \(y = -2(x - (-4))^2 + (-1)\)So \(h = -4,\ k = -1\). Vertex = \((-4, -1)\). The parabola opens downward (since \(a = -2 < 0\)). ✅ Answer: B
Q 09
Key insight: If \(b^m = b^n\), then \(m = n\). No calculator needed — just rewrite both sides with the same base!
Solve: \(4^{x+1} = 8^{x-1}\)
Solve: \(4^{x+1} = 8^{x-1}\)
Hint: Both 4 and 8 are powers of...
\(4 = 2^2\) and \(8 = 2^3\). Rewrite everything in base 2, then set exponents equal.
📖 Explanation
\(2^{2(x+1)} = 2^{3(x-1)}\)\(2x + 2 = 3x - 3\)
\(x = 5\) ✅ Answer: B
Q 10
Biggest mistake: Students cancel incorrectly, e.g., \(\dfrac{x+3}{x+3} \ne \dfrac{1}{1}\) when there's more going on. Always fully factor before cancelling.
Simplify: \(\dfrac{x^2 - 9}{x^2 - x - 6}\)
Simplify: \(\dfrac{x^2 - 9}{x^2 - x - 6}\)
📖 Explanation
Numerator: \(x^2 - 9 = (x+3)(x-3)\)Denominator: \(x^2 - x - 6 = (x-3)(x+2)\)
Cancel \((x-3)\): \(\dfrac{(x+3)\cancel{(x-3)}}{\cancel{(x-3)}(x+2)} = \dfrac{x+3}{x+2}\), where \(x \ne 3\) ✅ Answer: A
Geometry
Triangles · Circles · Area · Proofs · Coordinate Geometry
G 01
Watch out: "Hypotenuse" is always the side OPPOSITE the right angle — not just the longest-looking side. Identify the right angle first!
A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?
A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?
📖 Explanation
\(c^2 = 7^2 + 24^2 = 49 + 576 = 625\)\(c = \sqrt{625} = 25\)
This is a classic Pythagorean triple: \(7\text{-}24\text{-}25\)! ✅ Answer: A
G 02
Tricky: Students often forget the formula uses \((n-2)\) not just \(n\). Think of it this way: any polygon can be cut into \((n-2)\) triangles.
What is the measure of ONE interior angle of a regular hexagon?
What is the measure of ONE interior angle of a regular hexagon?
📖 Explanation
Sum of interior angles of hexagon (\(n=6\)):\((6-2) \times 180° = 4 \times 180° = 720°\)
Each angle in a regular hexagon: \(\dfrac{720°}{6} = 120°\) ✅ Answer: B
G 03
Common mix-up: Arc LENGTH uses the circumference formula \(2\pi r\). Arc AREA uses the area formula \(\pi r^2\). Don't swap them!
A circle has radius 9. Find the arc length cut off by a central angle of \(80°\). Leave your answer in terms of \(\pi\).
A circle has radius 9. Find the arc length cut off by a central angle of \(80°\). Leave your answer in terms of \(\pi\).
📖 Explanation
Arc length \(= \dfrac{80}{360} \times 2\pi(9) = \dfrac{2}{9} \times 18\pi = 4\pi\) ✅ Answer: B
G 04
Key trap: Make sure you match the CORRESPONDING sides! If the triangles are labeled differently, wrong side-pairing = wrong answer.
Triangles \(ABC\) and \(DEF\) are similar. If \(AB = 6\), \(BC = 9\), and \(DE = 10\), find \(EF\).
Triangles \(ABC\) and \(DEF\) are similar. If \(AB = 6\), \(BC = 9\), and \(DE = 10\), find \(EF\).
📖 Explanation
Set up proportion: \(\dfrac{AB}{DE} = \dfrac{BC}{EF}\)\(\dfrac{6}{10} = \dfrac{9}{EF}\)
\(EF = \dfrac{9 \times 10}{6} = 15\) ✅ Answer: C
G 05
Sneaky question type: They give you ONE endpoint and the midpoint, and ask for the OTHER endpoint. Use the formula backwards — algebra trick!
The midpoint of segment \(\overline{PQ}\) is \(M(3, -1)\). If \(P = (7, 5)\), find \(Q\).
The midpoint of segment \(\overline{PQ}\) is \(M(3, -1)\). If \(P = (7, 5)\), find \(Q\).
📖 Explanation
\(\dfrac{7 + x_Q}{2} = 3 \Rightarrow x_Q = -1\)\(\dfrac{5 + y_Q}{2} = -1 \Rightarrow y_Q = -7\)
So \(Q = (-1, -7)\) ✅ Answer: A
G 06
Don't forget the 1/3! Many students write \(\pi r^2 h\) (that's a cylinder). A cone holds exactly one-third the volume of a cylinder with the same base and height.
Find the volume of a cone with radius 6 and height 10. Leave in terms of \(\pi\).
Find the volume of a cone with radius 6 and height 10. Leave in terms of \(\pi\).
📖 Explanation
\(V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (6)^2 (10) = \dfrac{1}{3}\pi (36)(10) = \dfrac{360\pi}{3} = 120\pi\) ✅ Answer: A
G 07
Most confusing: Central angle = arc measure. Inscribed angle = HALF the arc. Students often flip these or use equal values.
An inscribed angle in a circle intercepts an arc of \(140°\). What is the measure of the inscribed angle?
An inscribed angle in a circle intercepts an arc of \(140°\). What is the measure of the inscribed angle?
Remember the Relationship
Central angle = arc = 140°. Inscribed angle (same arc) = 140° ÷ 2. This is the Inscribed Angle Theorem.
📖 Explanation
Inscribed Angle Theorem: Inscribed angle = \(\dfrac{1}{2}\) × intercepted arc\(= \dfrac{1}{2} \times 140° = 70°\) ✅ Answer: C
G 08
Classic trick: SSA (two sides and a non-included angle) does NOT guarantee congruence — this is called the "ambiguous case." AAA only proves similarity, not congruence.
Two triangles have two angles of \(45°\) and \(80°\) in common, and the side BETWEEN those angles is equal in both. Which congruence rule applies?
Two triangles have two angles of \(45°\) and \(80°\) in common, and the side BETWEEN those angles is equal in both. Which congruence rule applies?
📖 Explanation
Two angles and the included side (the side between the two known angles) = ASA (Angle-Side-Angle). The key word here is "between." ✅ Answer: C
G 09
Easy to mess up: It doesn't matter which point you call \((x_1, y_1)\) — since you're squaring the difference, the sign doesn't matter. But don't forget to square root at the end!
Find the distance between points \(A(-3, 2)\) and \(B(5, -4)\).
Find the distance between points \(A(-3, 2)\) and \(B(5, -4)\).
📖 Explanation
\(d = \sqrt{(5-(-3))^2 + (-4-2)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\) ✅ Answer: A
G 10
Confusing direction: Students mix up which side is \(x\sqrt{3}\) and which is \(2x\). Remember: the hypotenuse is always \(2x\), and the longer leg is \(x\sqrt{3}\).
In a 30-60-90 triangle, the hypotenuse is 14. Find the length of the shorter leg.
In a 30-60-90 triangle, the hypotenuse is 14. Find the length of the shorter leg.
30-60-90 Side Ratios
Short leg (30°) : Long leg (60°) : Hypotenuse = \(x\) : \(x\sqrt{3}\) : \(2x\)
If hypotenuse = 14, then \(2x = 14 \Rightarrow x = ?\)
If hypotenuse = 14, then \(2x = 14 \Rightarrow x = ?\)
📖 Explanation
Hypotenuse \(= 2x = 14 \Rightarrow x = 7\)The shorter leg (opposite 30°) \(= x = 7\)
The longer leg (opposite 60°) \(= x\sqrt{3} = 7\sqrt{3}\) ✅ Answer: C
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