📐
MY STUDY NOTEBOOK

Algebra 2 & Geometry

✦ Core Problems · Memory Keywords · Instant Feedback ✦

20 hand-picked problems students miss most. Each has a 📌 Memory Keyword, a worked example, and shows you the solution when you are wrong.

Name           
Date           
Score        
Score: 0 / 0
📘 Algebra 2 — Core Problems
Quadratics · Complex Numbers · Functions · Logarithms · Polynomials · Sequences · Systems
1
Quadratic Formula★★☆☆☆
📌 MEMORIZE: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ — "Neg-b, plus-or-minus, root of b-squared-minus-4ac, over 2a"
📖 Worked Example
$x^2-5x+6=0$, $a=1,b=-5,c=6$.  $x=\frac{5\pm\sqrt{25-24}}{2}=\frac{5\pm1}{2}$ so $x=3$ or $x=2$ ✔
Solve: $\;2x^2+3x-2=0$   ⚠ Watch the signs in the discriminant!
2
Discriminant★★☆☆☆
📌 DISC = $b^2-4ac$:  Positive→2 real  |  Zero→1 double root  |  Negative→No real (complex!)
How many real solutions does $3x^2-6x+3=0$ have?
3
Complex Numbers★★★☆☆
📌 CYCLE of $i$: $i^1=i,\;i^2=-1,\;i^3=-i,\;i^4=1$ — repeats every 4! Use remainder when dividing by 4.
📖 Example
$i^{13}$: $13\div4$ → remainder $1$, so $i^{13}=i^1=i$.
Simplify: $\;(3+2i)(1-4i)$   ⚠ Remember $i^2=-1$, not $+1$!
4
Function Composition★★☆☆☆
📌 INSIDE OUT! $f(g(x))$: do $g$ FIRST, then plug that result into $f$. Order matters — $f(g)\neq g(f)$!
Let $f(x)=2x+1$ and $g(x)=x^2$.  Find $\;f\!\left(g(3)\right)$.
5
Logarithm Properties★★★☆☆
📌 LOG RULES:  $\log(AB)=\log A+\log B$ ("multiply=ADD")  |  $\log(A/B)=\log A-\log B$ ("divide=SUB")  |  $\log(A^n)=n\log A$ ("power=FRONT")
Simplify: $\;\log_2 8+\log_2 4$
★ keep going! ★
6
Inverse Functions★★☆☆☆
📌 SWAP $x$ and $y$! Write $y=f(x)$, then swap $x\leftrightarrow y$, then solve for $y$. That's your inverse.
Find the inverse of $f(x)=3x-6$.
7
Factor Theorem★★★☆☆
📌 FACTOR THEOREM: If $f(c)=0$ then $(x-c)$ IS a factor. Just PLUG IN to test.
$f(x)=x^3-6x^2+11x-6$. Which value is NOT a root?
8
Exponential Equations★★★☆☆
📌 SAME BASE → SET EXPONENTS EQUAL! If $a^m=a^n$ and $a\neq1$, then $m=n$.
Solve: $\;4^{x+1}=8^x$   ⚠ Hint: rewrite both with base 2!
9
Arithmetic Sequences★★☆☆☆
📌 $a_n=a_1+(n-1)d$ — "first term + (number of steps × common difference)"
The 4th term of an arithmetic sequence is $10$ and $d=3$. Find the 10th term.
10
Linear-Quadratic System★★★☆☆
📌 SUBSTITUTE! Plug the linear equation into the quadratic. Then simplify, factor, solve.
📖 Example
$y=x+2$ and $y=x^2$. Substitute: $x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow(x-2)(x+1)=0$.
Find all $x$: $\;y=x^2-1\;$ and $\;y=x+1$.
📐 Geometry — Core Problems
Triangles · Circles · Volume · Similarity · Angle Pairs · Coordinate Geometry
1
Pythagorean Theorem★☆☆☆☆
📌 $a^2+b^2=c^2$ — $c$ is ALWAYS the hypotenuse (longest side). Common triples: 3-4-5, 5-12-13, 8-15-17
A right triangle has legs $6$ and $8$. What is the hypotenuse?
2
Special Right Triangles★★★☆☆
📌 45-45-90: $x:x:x\sqrt{2}$ ("legs equal, hyp = leg × √2")  |  30-60-90: $x:x\sqrt{3}:2x$ ("short : ×√3 : double")
In a 30-60-90 triangle, the hypotenuse is 10. What is the shorter leg?
3
Arc Length★★☆☆☆
📌 ARC LENGTH $=\dfrac{\theta}{360°}\times2\pi r$ — "fraction of the full circumference"
Circle: radius $9$, central angle $120°$. Find the arc length. (Leave in terms of $\pi$.)
4
Similar Triangles (AA)★★★☆☆
📌 AA SIMILARITY: Two equal angles → triangles similar → all sides in the SAME ratio. Set up: $\frac{\text{small side}}{\text{big side}}=\frac{\text{small perimeter}}{\text{big perimeter}}$
Two similar triangles have sides in ratio $2:5$. The smaller has perimeter $18$. What is the larger triangle's perimeter?
5
Volume of a Cone★★☆☆☆
📌 CONE = $\frac{1}{3}$ of CYLINDER: $V=\dfrac{1}{3}\pi r^2 h$ — "A cone fills its cylinder exactly one-third."
A cone has radius $3$ and height $7$. Find its volume. (Leave in terms of $\pi$.)
★ halfway there! ★
6
Parallel Lines & Transversals★★☆☆☆
📌 SHAPES:  Z → Alternate interior → EQUAL  |  C → Co-interior → ADD to 180°  |  F → Corresponding → EQUAL
Two parallel lines cut by a transversal. One co-interior angle is $72°$. Find the other co-interior angle.
7
Inscribed Angle Theorem★★★☆☆
📌 INSCRIBED = HALF of CENTRAL (same arc). If central = 80°, inscribed = 40°. "Half is inscribed, Full is central."
A central angle intercepts an arc of $140°$. What is the inscribed angle intercepting the same arc?
8
Midpoint Formula★☆☆☆☆
📌 MIDPOINT = AVERAGE of each coordinate: $M=\!\left(\dfrac{x_1+x_2}{2},\;\dfrac{y_1+y_2}{2}\right)$
Endpoints: $A(2,-4)$ and $B(8,6)$. Find the midpoint.
9
Exterior Angle Theorem★★★☆☆
📌 EXTERIOR ANGLE = sum of the TWO REMOTE interior angles. Do NOT include the adjacent angle — just the two NOT next to it!
In triangle $ABC$, exterior angle at $C$ is $125°$. If $\angle A=60°$, find $\angle B$.
10
Circle Equation (Standard Form)★★★☆☆
📌 $(x-h)^2+(y-k)^2=r^2$ → center=$(h,k)$, radius=$r$.  ⚠ SIGNS FLIP! $(x+3)^2$ means $h=-3$, not $+3$!
📖 Example
$(x-2)^2+(y+1)^2=25$ → Center: $(2,-1)$, Radius: $5$.
Find center and radius: $\;(x+3)^2+(y-4)^2=49$