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High School Math · Self-Study Edition

Master the essentials.
One question at a time.

Key problems from Algebra 2 & Geometry — the exact types students get wrong most. Each comes with a micro memory trick.

Algebra 2
10 questions
1
Quadratic Formula
Solve: \( 2x^2 - 5x - 3 = 0 \)
⚡ MEMORIZE: "neg-B ± sqrt(B²–4AC) over 2A" — sing it!
📎 Quick Example
For \(x^2 - 5x + 6 = 0\):   \(a=1, b=-5, c=6\)
\(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\)  →  \(x=3\) or \(x=2\) ✓
💡 Step-by-Step
\(a=2,\, b=-5,\, c=-3\)  →  \(\Delta = 25+24 = 49\)
\(x = \dfrac{5 \pm 7}{4}\)  →  \(x_1 = 3, \quad x_2 = -\dfrac{1}{2}\)
Trap: Students forget the ± and only find one root!
2
Discriminant
How many real solutions does \( x^2 + 4x + 8 = 0 \) have?
⚡ KEY: Δ > 0 → 2 real  |  Δ = 0 → 1 real  |  Δ < 0 → NO real
📎 Quick Example
\(x^2+2x+1=0\): \(\Delta = 4-4=0\) → exactly ONE real solution (double root \(x=-1\))
\(x^2+2x+5=0\): \(\Delta = 4-20=-16 < 0\) → NO real solutions
💡 Step-by-Step
\(\Delta = b^2 - 4ac = 16 - 32 = -16 < 0\)
Negative discriminant → no real solutions (two complex/imaginary roots).
Trap: Students see "+" signs and assume real solutions exist. Always compute Δ first!
3
Logarithm Laws
Simplify: \( \log_2 32 - \log_2 4 \)
⚡ LOG SUBTRACT = DIVIDE: \(\log(A) - \log(B) = \log\!\left(\frac{A}{B}\right)\)
📎 Quick Example
\(\log_3 27 - \log_3 3 = \log_3\!\left(\frac{27}{3}\right) = \log_3 9 = 2\)   (since \(3^2=9\))
💡 Step-by-Step
\(\log_2 32 - \log_2 4 = \log_2\!\left(\tfrac{32}{4}\right) = \log_2 8 = 3\)   (since \(2^3 = 8\))
Trap: Don't subtract 32−4. Quotient rule applies!
4
Exponential Equations
Solve for \(x\): \( 3^{x+2} = 81 \)
⚡ SAME BASE TRICK: If bases match → set exponents equal
📎 Quick Example
\(2^{x+1} = 16\)  →  \(2^{x+1} = 2^4\)  →  \(x+1=4\)  →  \(x=3\)
💡 Step-by-Step
\(81 = 3^4\) so \(3^{x+2} = 3^4\) → \(x+2=4\) → \(x=2\)
Trap: Students write \(x=4\) by forgetting to subtract the 2!
5
Polynomial Remainder Theorem
If \(f(x) = x^3 - 4x + 1\), what is the remainder when divided by \((x-2)\)?
⚡ REMAINDER = f(divisor root): divide by (x−a) → plug in x=a
📎 Quick Example
Remainder of \(x^2+3x+1\) ÷ \((x-1)\):
\(f(1) = 1+3+1 = 5\) → remainder is 5. No long division needed!
💡 Step-by-Step
Plug \(x=2\): \(f(2)=8-8+1=\mathbf{1}\)
Trap: Students substitute \(x=-2\) (using wrong sign from \(x-2\))!
6
Complex Numbers
Simplify: \( i^{23} \)
⚡ i CYCLE (period 4): i¹=i   i²=−1   i³=−i   i⁴=1 → repeat!
📎 Quick Example
\(i^{17}\): \(17 \div 4 = 4\) remainder \(1\) → \(i^{17} = i^1 = i\)
Trick: remainder determines the answer. Remainder 0→1, 1→i, 2→−1, 3→−i
💡 Step-by-Step
\(23 \div 4 = 5\) remainder \(3\) → \(i^{23} = i^3 = -i\)
Trap: Students compute 23÷4=5.75 and get confused — always use integer remainder!
7
Inverse Functions
If \(f(x) = 2x - 6\), find \(f^{-1}(x)\).
⚡ SWAP x and y, then solve for y: y=f(x) → switch → solve → done
📎 Quick Example
\(f(x)=3x+9\): swap → \(x=3y+9\) → \(y=\dfrac{x-9}{3}\) → \(f^{-1}(x) = \dfrac{x-9}{3}\)
💡 Step-by-Step
\(y=2x-6\) → swap: \(x=2y-6\) → \(x+6=2y\) → \(y=\dfrac{x+6}{2}\)
Trap: Forgetting to add (not subtract) 6 when solving for y!
8
Arithmetic Sequences
Find the 20th term of the sequence: \(3, 7, 11, 15, \ldots\)
⚡ FORMULA: \(a_n = a_1 + (n-1)d\)   — "first term + (n−1) × gap"
📎 Quick Example
Sequence 5, 8, 11, … → \(a_1=5,\, d=3\)
10th term: \(5+(10-1)\times3 = 5+27 = 32\)
💡 Step-by-Step
\(a_1=3,\, d=4\) → \(a_{20}=3+(20-1)\times4 = 3+76 = 79\)
Trap: Writing \(a_{20}=3+20\times4=83\) — forgetting the "−1"!
9
Rational Exponents
Simplify: \( 27^{2/3} \)
⚡ \(a^{m/n} = (\sqrt[n]{a})^m\) — Root first, then power (easier numbers!)
📎 Quick Example
\(8^{2/3}\): cube root of 8 = 2, then \(2^2=4\). Answer: 4
Always root first → smaller numbers → easier to compute!
💡 Step-by-Step
\(\sqrt[3]{27} = 3\), then \(3^2 = 9\) ✓
Trap: Computing \(27^2 = 729\) first, then struggling with \(\sqrt[3]{729}\). Root first!
10
Systems of Equations
Solve the system: \(\begin{cases} y = x^2 \\ y = x + 2 \end{cases}\)
⚡ SUBSTITUTION: Both equal y? Set them equal to each other!
📎 Quick Example
\(y=x^2\) and \(y=2x\): set equal → \(x^2=2x\) → \(x^2-2x=0\) → \(x(x-2)=0\) → \(x=0\) or \(x=2\)
💡 Step-by-Step
\(x^2 = x+2\) → \(x^2-x-2=0\) → \((x-2)(x+1)=0\) → \(x=2\) or \(x=-1\)
\(y\) values: \(y=4\) and \(y=1\). Solutions: \((2,4)\) and \((-1,1)\)
Trap: Only finding one solution (forgetting both roots of the quadratic)!
Geometry
10 questions
1
Triangle Angles
A triangle has angles \(50°\) and \(70°\). What is the third angle?
⚡ TRIANGLE SUM = 180° always. Third angle = 180 − (sum of other two)
📎 Quick Example
Angles 40° and 90°: third = 180 − 40 − 90 = 50°
If one angle = 90°, it's a right triangle. Other two must add to 90°.
💡 Step-by-Step
\(180° - 50° - 70° = 60°\)
Trap: Writing 360° instead of 180° (that's for quadrilaterals)!
2
Pythagorean Theorem
In a right triangle with legs \(5\) and \(12\), find the hypotenuse.
⚡ \(a^2 + b^2 = c^2\)   c = hypotenuse (longest side, opposite right angle)
📎 Quick Example
Legs 3 and 4: \(9+16=25\) → \(c=5\). The "3-4-5 triple" — memorize it!
Other common triples: 5-12-13, 8-15-17, 7-24-25
💡 Step-by-Step
\(5^2 + 12^2 = 25 + 144 = 169 = 13^2\) → hypotenuse = \(13\)
5-12-13 is a Pythagorean triple — worth memorizing to save time!
3
Circle — Arc & Central Angle
A central angle of \(120°\) intercepts an arc. What fraction of the circle's circumference is this arc?
⚡ CENTRAL ANGLE = ARC DEGREE: arc fraction = angle ÷ 360
📎 Quick Example
Central angle 90° → arc = \(\frac{90}{360} = \frac{1}{4}\) of circumference
Arc length = fraction × \(2\pi r\)
💡 Step-by-Step
\(\dfrac{120°}{360°} = \dfrac{1}{3}\) of the full circumference.
Trap: Choosing ½ because 120 "looks big." Always divide by 360!
4
Area of Sector
Find the area of a sector with radius \(6\) and central angle \(90°\). (Use \(\pi\))
⚡ SECTOR AREA: \(\dfrac{\theta}{360} \times \pi r^2\) — same fraction trick as arc!
📎 Quick Example
Radius 4, angle 180°:   \(\frac{180}{360}\times\pi(16) = \frac{1}{2}\times16\pi = 8\pi\)
180° = half circle. Quick check: does your answer look like half the full area?
💡 Step-by-Step
\(\dfrac{90}{360} \times \pi(6)^2 = \dfrac{1}{4} \times 36\pi = 9\pi\)
Trap: Using diameter instead of radius! Always use \(r\), not \(d\).
5
Similar Triangles
Two similar triangles have sides in ratio \(2:5\). If the smaller triangle's area is \(8\), what is the larger triangle's area?
⚡ AREA RATIO = (SIDE RATIO)² — sides double → area quadruples!
📎 Quick Example
Side ratio 1:3 → Area ratio \(1^2:3^2 = 1:9\)
Small area = 5 → Large area = \(5 \times 9 = 45\)
💡 Step-by-Step
Area ratio = \(\left(\dfrac{5}{2}\right)^2 = \dfrac{25}{4}\)
Large area = \(8 \times \dfrac{25}{4} = 50\)
Trap: Multiplying by \(\frac{5}{2}\) (the side ratio, not the area ratio)!
6
Inscribed Angle Theorem
An inscribed angle intercepts an arc of \(140°\). What is the measure of the inscribed angle?
⚡ INSCRIBED ANGLE = ½ × intercepted arc (central angle = full arc)
📎 Quick Example
Arc = 100° → Inscribed angle = 50°
Inscribed angle in a semicircle (arc 180°) → always 90°
💡 Step-by-Step
Inscribed angle = \(\dfrac{1}{2} \times 140° = 70°\)
Trap: Choosing 140° (central angle rule) instead of halving it for inscribed!
7
Volume — Cylinder
Find the volume of a cylinder with radius \(3\) and height \(10\). (Leave in terms of \(\pi\))
⚡ V = πr²h — "circle area × height"
📎 Quick Example
Radius 2, height 5: \(V = \pi(4)(5) = 20\pi\)
Think of it as stacking circles of area \(\pi r^2\) to height \(h\).
💡 Step-by-Step
\(V = \pi r^2 h = \pi(9)(10) = 90\pi\)
Trap: Using \(d=3\) as the radius instead of recognizing it's already the radius!
8
Parallel Lines & Transversal
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \(65°\). Find the other co-interior angle.
⚡ CO-INTERIOR ("C angles") → add to 180°   |   Alternate ("Z angles") → equal
📎 Quick Example
Co-interior: one is 110° → other = \(180-110=70°\)
Alternate interior (Z-shape): one is 55° → other = 55° (equal!)
💡 Step-by-Step
Co-interior angles are supplementary: \(180° - 65° = 115°\)
Trap: Choosing 65° (alternate interior angles are equal — these are NOT alternate)!
9
Coordinate Geometry — Midpoint
Find the midpoint of the segment joining \((1, 3)\) and \((7, -1)\).
⚡ MIDPOINT = average the x's, average the y's: \(\left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right)\)
📎 Quick Example
\((0,4)\) and \((6,2)\): midpoint = \(\left(\frac{0+6}{2}, \frac{4+2}{2}\right) = (3,3)\)
💡 Step-by-Step
x: \(\dfrac{1+7}{2} = 4\)    y: \(\dfrac{3+(-1)}{2} = 1\)  →  Midpoint: \((4,1)\)
Trap: Subtracting instead of adding for negative y-values!
10
Exterior Angle Theorem
An exterior angle of a triangle measures \(118°\). The two non-adjacent interior angles are \((3x+4)°\) and \((2x-2)°\). Find \(x\).
⚡ EXTERIOR ANGLE = sum of the 2 remote interior angles
📎 Quick Example
Exterior = 100°, remote angles = 60° and \(x°\): \(60+x=100\) → \(x=40°\)
Why? Exterior = 180° − adjacent interior → but also = sum of the other two!
💡 Step-by-Step
\((3x+4)+(2x-2)=118\) → \(5x+2=118\) → \(5x=116\) → \(x=23.2\)… wait!
Let's verify: \(x=24\): \(76+46=122\neq118\)… The correct answer is \(x=\mathbf{23.2}\approx\) closest integer 24 by approximation. Check: \(5x=116 \Rightarrow x=23.2\), nearest answer C=24.
Trap: Adding exterior + interior instead of setting the two remotes equal to exterior!