📓 MATH NOTEBOOK

Algebra 1 & Geometry · Self-Study Edition
✨ 20 Key Problems with Memory Tips
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📘 Algebra 1

Questions 1–10 · Core Topics · Fill in your notes below!
Q1 · Solving Linear Equations
One-Step & Two-Step Equations
INVERSE OPERATIONS = "Undo it backwards"
+/− ↔ −/+ · × ↔ ÷ (do to BOTH sides!)
📝 EXAMPLE
Solve: \(2x + 5 = 13\)
Step 1: \(2x = 13 - 5 = 8\)
Step 2: \(x = \dfrac{8}{2} = 4\) ✔
Solve for \(x\):
\(3x - 7 = 14\)
💡 EXPLANATION
\(3x - 7 = 14\) → Add 7 to both sides: \(3x = 21\) → Divide by 3: \(x = 7\)
Trap alert: Students forget to add 7 first before dividing! Always "undo" addition/subtraction BEFORE multiplication/division.
Q2 · Slope-Intercept Form
Reading \(y = mx + b\)
\(y = mx + b\) : m = SLOPE (steepness), b = Y-INTERCEPT (start point)
"My Bagel" → m = slope, b = where line crosses y-axis
📝 EXAMPLE
Line: \(y = -2x + 5\) → Slope \(= -2\), y-intercept \(= (0, 5)\)
Which line has a slope of \(3\) and passes through \((0, -4)\)?
💡 EXPLANATION
\(m = 3\), \(b = -4\) → plug into \(y = mx + b\): \(y = 3x + (-4) = 3x - 4\)
Trap alert: Don't mix up m and b positions! m always multiplies x, b is the constant.
Q3 · Systems of Equations
Substitution Method
SUBSTITUTION: Solve one → PLUG IN the other
"Swap & Solve" — replace one variable with an expression
📝 EXAMPLE
\(y = 2x\) and \(x + y = 9\)
→ Sub: \(x + 2x = 9\) → \(3x = 9\) → \(x = 3, y = 6\)
Solve the system: \(\begin{cases} y = x + 3 \\ 2x + y = 12 \end{cases}\)
What is \(x\)?
💡 EXPLANATION
Sub \(y = x+3\) into equation 2: \(2x + (x+3) = 12\) → \(3x + 3 = 12\) → \(3x = 9\) → \(\mathbf{x = 3}\)
Then \(y = 3 + 3 = 6\). Check: \(2(3) + 6 = 12\) ✔
Q4 · Inequalities
Solving & Graphing Inequalities
FLIP THE SIGN when you × or ÷ by a NEGATIVE number!
"Multiply negative → Arrow flips direction"
📝 EXAMPLE
\(-2x > 8\) → Divide by \(-2\) (FLIP!): \(x < -4\)
Solve: \(-3x + 6 \geq 15\)
Which inequality is correct?
💡 EXPLANATION
\(-3x + 6 \geq 15\) → \(-3x \geq 9\) → divide by \(-3\) (FLIP \(\geq\) to \(\leq\)): \(x \leq -3\)
Trap alert: Most students forget to FLIP the inequality sign when dividing by negative!
Q5 · Exponent Rules
Laws of Exponents
SAME BASE: Multiply → ADD exponents · Divide → SUBTRACT exponents
"Same base? Just ADD or SUBTRACT the powers"
📝 EXAMPLE
\(x^3 \cdot x^4 = x^{3+4} = x^7\)
\(\dfrac{x^6}{x^2} = x^{6-2} = x^4\)
Simplify: \(\dfrac{a^5 \cdot a^3}{a^4}\)
💡 EXPLANATION
Numerator: \(a^5 \cdot a^3 = a^{5+3} = a^8\)
Then: \(\dfrac{a^8}{a^4} = a^{8-4} = a^4\)
Trap alert: Don't multiply 5×3 or subtract wrong. Always ADD first, then SUBTRACT.
Q6 · Factoring
Factoring Quadratics \(ax^2 + bx + c\)
Find two numbers: MULTIPLY to \(c\), ADD to \(b\)
"Product = c, Sum = b → Find the pair!"
📝 EXAMPLE
\(x^2 + 5x + 6\): need × = 6, + = 5 → \(2 \times 3 = 6\), \(2+3=5\)
Answer: \((x+2)(x+3)\)
Factor completely: \(x^2 - x - 12\)
💡 EXPLANATION
Need × = \(-12\), + = \(-1\)
Try: \(-4 \times 3 = -12\) ✔ and \(-4 + 3 = -1\) ✔
So: \((x-4)(x+3)\)
Trap alert: Watch the SIGNS carefully! Many students use +4 and -3 which gives sum of +1, not -1.
Q7 · Quadratic Formula
Using \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
"Negative B, Plus or Minus, Square Root, B² minus 4AC, All over 2A"
DISCRIMINANT \(b^2 - 4ac\): >0 two roots · =0 one root · <0 no real roots
📝 EXAMPLE
\(x^2 - 5x + 6 = 0\), \(a=1, b=-5, c=6\)
\(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\) → \(x=3\) or \(x=2\)
How many real solutions does \(x^2 + 4x + 5 = 0\) have?
💡 EXPLANATION
Discriminant: \(b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4\)
Since \(-4 < 0\), there are NO real solutions (square root of negative is imaginary).
Trap alert: Students often compute \(16 - 4 = 12\) (forgetting to multiply by \(a\) and \(c\)).
Q8 · Functions
Function Notation & Evaluation
\(f(x)\) means "output when input is x" — just SUBSTITUTE!
"f(3) → replace every x with 3"
📝 EXAMPLE
\(f(x) = 2x^2 - 1\)
\(f(3) = 2(3)^2 - 1 = 18 - 1 = 17\)
If \(f(x) = x^2 - 3x + 2\), what is \(f(-2)\)?
💡 EXPLANATION
\(f(-2) = (-2)^2 - 3(-2) + 2 = 4 + 6 + 2 = 12\)
Trap alert: \((-2)^2 = +4\) (NOT \(-4\)!) · \(-3 \times (-2) = +6\) (negative × negative = positive!)
Q9 · Polynomials
FOIL Method — Multiplying Binomials
FOIL = First · Outer · Inner · Last
\((a+b)(c+d) = ac + ad + bc + bd\)
📝 EXAMPLE
\((x+3)(x-2)\)
F: \(x^2\) · O: \(-2x\) · I: \(3x\) · L: \(-6\)
\(= x^2 + x - 6\)
Expand: \((2x - 3)(x + 5)\)
💡 EXPLANATION
F: \(2x \cdot x = 2x^2\) · O: \(2x \cdot 5 = 10x\) · I: \(-3 \cdot x = -3x\) · L: \(-3 \cdot 5 = -15\)
Combine: \(2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15\)
Trap alert: Don't forget the OUTER and INNER terms — students often only multiply First and Last!
Q10 · Absolute Value
Absolute Value Equations
\(|x| = k\) → TWO equations: \(x = k\) OR \(x = -k\)
"Absolute = Always positive → Split into TWO cases!"
📝 EXAMPLE
\(|x - 3| = 5\)
Case 1: \(x - 3 = 5 \Rightarrow x = 8\)
Case 2: \(x - 3 = -5 \Rightarrow x = -2\)
Solve: \(|2x + 1| = 9\)
What are the solutions?
💡 EXPLANATION
Case 1: \(2x+1 = 9 \Rightarrow 2x = 8 \Rightarrow x = 4\)
Case 2: \(2x+1 = -9 \Rightarrow 2x = -10 \Rightarrow x = -5\)
Trap alert: Students only solve one case! Absolute value ALWAYS gives two equations.

📐 Geometry

Questions 11–20 · Shapes, Angles, Proofs · Write your work below!
Q11 · Angle Relationships
Complementary & Supplementary Angles
COMPLEMENTARY = adds to 90° (think: "C" comes before "S" like 90 before 180)
SUPPLEMENTARY = adds to 180° · "S = Straight line = 180°"
📝 EXAMPLE
If \(\angle A = 35°\), its complement \(= 90° - 35° = 55°\)
Its supplement \(= 180° - 35° = 145°\)
Two angles are supplementary. One angle is \(3x + 10\) degrees and the other is \(2x - 20\) degrees.
Find \(x\).
💡 EXPLANATION
Supplementary → sum = 180°
\((3x+10) + (2x-20) = 180\) → \(5x - 10 = 180\) → \(5x = 190\) → \(x = 38\)
Check: \(3(38)+10 = 124°\), \(2(38)-20 = 56°\), \(124+56 = 180°\) ✔
Q12 · Parallel Lines & Transversals
Alternate Interior Angles
ALTERNATE INTERIOR = Equal (Z-shape between lines)
CO-INTERIOR (same side) = Add to 180° · "Z = equal, C = 180°"
📝 EXAMPLE
Parallel lines cut by a transversal:
Alternate interior angles are EQUAL · Corresponding angles are EQUAL
Two parallel lines are cut by a transversal. One alternate interior angle is \((4x + 15)°\) and the other is \((7x - 12)°\).
Find \(x\).
💡 EXPLANATION
Alternate interior angles are EQUAL:
\(4x + 15 = 7x - 12\) → \(27 = 3x\) → \(x = 9\)
Check: \(4(9)+15 = 51°\), \(7(9)-12 = 51°\) ✔
Q13 · Triangle Basics
Triangle Angle Sum Theorem
All 3 angles of ANY triangle add to 180°
"3 corners, always 180° — no exceptions!"
📝 EXAMPLE
Triangle with angles \(50°\) and \(70°\):
Third angle \(= 180° - 50° - 70° = 60°\)
A triangle has angles \((2x)°\), \((x + 30)°\), and \((3x - 10)°\).
What is the largest angle?
💡 EXPLANATION
\(2x + (x+30) + (3x-10) = 180\) → \(6x + 20 = 180\) → \(x = \frac{160}{6} \approx 26.67°\)
Wait — let's recheck: \(6x = 160\) → \(x = \frac{160}{6}\)? Let me use whole numbers: \(6x + 20 = 180 \Rightarrow 6x = 160 \Rightarrow x = \frac{80}{3}\)
Largest angle: \(3x - 10 = 3(\frac{80}{3}) - 10 = 80 - 10 = \mathbf{70°}\). Wait — angles: \(2x \approx 53.3°\), \(x+30 \approx 56.7°\), \(3x-10 = 70°\).
Actually x = 160/6 = 26.67, largest = 3(26.67)-10 = 70°. Answer: B) 70°
Trap: Sum ALL three before solving. Don't find x and stop — plug back in!
Q14 · Pythagorean Theorem
\(a^2 + b^2 = c^2\) (Right Triangles)
\(a^2 + b^2 = c^2\) · c is always the HYPOTENUSE (longest side, opposite right angle)
Common triples: 3-4-5 · 5-12-13 · 8-15-17
📝 EXAMPLE
Legs = 6 and 8 → \(6^2 + 8^2 = 36 + 64 = 100\) → \(c = \sqrt{100} = 10\)
A right triangle has legs of length \(9\) and \(12\). What is the hypotenuse?
💡 EXPLANATION
\(9^2 + 12^2 = 81 + 144 = 225\) → \(c = \sqrt{225} = 15\)
Note: B and C are equivalent! But the simplified answer is \(15\).
This is a 3-4-5 triple scaled by 3: \(9 = 3\times3\), \(12 = 4\times3\), \(15 = 5\times3\).
Q15 · Area & Perimeter
Area of Triangles & Parallelograms
Triangle: \(A = \frac{1}{2}bh\) · Parallelogram: \(A = bh\)
"Triangle = HALF of parallelogram with same base & height"
📝 EXAMPLE
Triangle: base = 10, height = 6
\(A = \frac{1}{2}(10)(6) = 30\) sq units
A parallelogram has base \(14\) cm and height \(9\) cm.
A triangle has the same base and height. What is the difference in their areas?
💡 EXPLANATION
Parallelogram: \(A = 14 \times 9 = 126 \text{ cm}^2\)
Triangle: \(A = \frac{1}{2}(14)(9) = 63 \text{ cm}^2\)
Difference: \(126 - 63 = 63 \text{ cm}^2\)
Q16 · Circles
Circumference & Area of Circles
Circumference: \(C = 2\pi r = \pi d\) · Area: \(A = \pi r^2\)
"Circles need \(\pi\) — r for Area (r²), d for Circumference (shortcut)"
📝 EXAMPLE
Circle with \(r = 5\):
\(C = 2\pi(5) = 10\pi \approx 31.4\) · \(A = \pi(5)^2 = 25\pi \approx 78.5\)
A circle has diameter \(d = 10\) cm. Which expression gives its area?
💡 EXPLANATION
\(d = 10\) → \(r = 5\) (HALF the diameter!)
\(A = \pi r^2 = \pi(5)^2 = 25\pi\)
Trap alert: Most students forget to halve the diameter! They use r=10 and get \(100\pi\) (Option A).
Q17 · Similar Triangles
Proportional Sides in Similar Triangles
SIMILAR = Same shape, different size → Corresponding sides are PROPORTIONAL
"Set up a fraction ratio: \(\frac{small}{big} = \frac{small}{big}\)"
📝 EXAMPLE
\(\triangle ABC \sim \triangle DEF\), \(AB = 4, DE = 8, BC = 6\)
\(\frac{4}{8} = \frac{6}{EF}\) → \(EF = 12\)
\(\triangle PQR \sim \triangle XYZ\). If \(PQ = 6\), \(XY = 9\), and \(QR = 10\), find \(YZ\).
💡 EXPLANATION
Set up proportion: \(\dfrac{PQ}{XY} = \dfrac{QR}{YZ}\)
\(\dfrac{6}{9} = \dfrac{10}{YZ}\) → \(6 \cdot YZ = 90\) → \(YZ = 15\)
Q18 · Volume
Volume of 3D Shapes
Prism/Cylinder: \(V = Bh\) (Base area × height)
Pyramid/Cone: \(V = \frac{1}{3}Bh\) — "Pointy = one-third of the box!"
📝 EXAMPLE
Cylinder: \(r=3, h=10\) → \(V = \pi(3)^2(10) = 90\pi\)
Cone: same → \(V = \frac{1}{3}(90\pi) = 30\pi\)
A rectangular prism has length \(8\), width \(5\), and height \(4\).
A pyramid has the same base and height. What is the pyramid's volume?
💡 EXPLANATION
Prism volume: \(8 \times 5 \times 4 = 160\)
Pyramid: \(\frac{1}{3}(8 \times 5)(4) = \frac{1}{3}(160) \approx 53.3 \text{ units}^3\)
Trap alert: Forgetting the \(\frac{1}{3}\) for pyramids! Prism and pyramid look similar but pyramid is exactly 1/3 the volume.
Q19 · Transformations
Reflections & Coordinate Rules
Reflect over x-axis: \((x, y) \rightarrow (x, -y)\) — flip the y!
Reflect over y-axis: \((x, y) \rightarrow (-x, y)\) — flip the x!
"x-axis flips y · y-axis flips x — opposite!"
📝 EXAMPLE
Point \(A(3, -5)\) reflected over x-axis → \(A'(3, 5)\)
Same point over y-axis → \(A'(-3, -5)\)
Point \(P(-4, 7)\) is reflected over the x-axis, then over the y-axis.
What are the final coordinates?
💡 EXPLANATION
Step 1 — Over x-axis: \((-4, 7) \rightarrow (-4, -7)\) (flip y)
Step 2 — Over y-axis: \((-4, -7) \rightarrow (4, -7)\) (flip x)
Final answer: \((4, -7)\)
Trap alert: Students mix up which coordinate to flip. Remember: reflect over x → negate y (the OTHER one)!
Q20 · Congruence
Triangle Congruence Theorems
SSS · SAS · ASA · AAS · HL (right triangles only)
"Shake Sides Angles: SSS, SAS, ASA, AAS — remember HL for RIGHT triangles!"
⚠️ SSA and AAA do NOT prove congruence!
📝 EXAMPLE
Two triangles share a side, with two equal angles on either side → ASA ✔
Two sides and the INCLUDED angle equal → SAS ✔
Two triangles have two pairs of equal sides and the included angles are equal.
Which congruence theorem applies?
💡 EXPLANATION
"Two pairs of equal sides AND the included angle" = Side - Angle - Side = SAS
The angle must be BETWEEN the two sides (included)!
Trap alert: If the angle is NOT between the sides, it's SSA — which does NOT prove congruence!