πŸ“ Math Study Notebook
Algebra 1 & Geometry Β· Self-Study Edition
✏️ πŸ“ πŸ”’ πŸ“ ✨
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πŸ”’ ALGEBRA 1 β€” Problems 1–10
1
πŸ“Œ ISOLATE x = do the OPPOSITE
Linear Equations Β· One-Step
✏️ EXAMPLE
Solve: x + 5 = 12
β†’ Subtract 5 from both sides β†’ x = 7
Solve for x: 3x βˆ’ 9 = 15
⚠️ Tricky!
Add 9 first, then divide by 3. Don't forget to add 9 to BOTH sides!
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2
πŸ“Œ DISTRIBUTE first β†’ then COMBINE like terms
Distributive Property & Combining Like Terms
✏️ EXAMPLE
Simplify: 2(x + 3) + 4x
β†’ 2x + 6 + 4x β†’ 6x + 6
Simplify: 3(2x βˆ’ 4) + 5x βˆ’ 2
⚠️ Tricky!
3 Γ— (βˆ’4) = βˆ’12, not +12. Watch the negative sign!
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3
πŸ“Œ VARIABLES on one side β†’ CONSTANTS on other
Two-Step & Multi-Step Equations
✏️ EXAMPLE
Solve: 5x + 3 = 2x + 12
β†’ Subtract 2x: 3x + 3 = 12 β†’ Subtract 3: 3x = 9 β†’ x = 3
Solve for x: 4x + 7 = 2x βˆ’ 3
⚠️ Negative Answer!
Move 2x to the left. You'll get a negative answer β€” that's okay!
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4
πŸ“Œ SLOPE = RISE Γ· RUN = (yβ‚‚βˆ’y₁)/(xβ‚‚βˆ’x₁)
Slope of a Line
✏️ EXAMPLE
Slope between (1, 2) and (3, 6):
m = (6βˆ’2)/(3βˆ’1) = 4/2 = 2
Find the slope of the line through (βˆ’2, 3) and (4, βˆ’9).
⚠️ Both Negative!
Carefully subtract: yβ‚‚ βˆ’ y₁ = βˆ’9 βˆ’ 3 = βˆ’12. Don't add them!
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5
πŸ“Œ y = mx + b Β· m = slope Β· b = y-intercept
Slope-Intercept Form
✏️ EXAMPLE
Line with slope 3 and y-intercept βˆ’1:
y = 3x βˆ’ 1
Which equation has slope βˆ’23 and passes through (0, 5)?
⚠️ Fraction slope!
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6
πŸ“Œ INEQUALITY: flip sign when DIVIDE/MULTIPLY by NEGATIVE
Solving Inequalities
✏️ EXAMPLE
Solve: βˆ’2x < 8
β†’ Divide by βˆ’2 β†’ FLIP! β†’ x > βˆ’4
Solve: βˆ’3x + 6 β‰₯ βˆ’9
⚠️ Flip the sign!
When you divide both sides by βˆ’3, the β‰₯ becomes ≀!
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7
πŸ“Œ FOIL = First Β· Outer Β· Inner Β· Last
Multiplying Polynomials (FOIL)
✏️ EXAMPLE
(x+2)(x+3)
F: xΒ·x = xΒ² Β· O: xΒ·3 = 3x Β· I: 2Β·x = 2x Β· L: 2Β·3 = 6
= xΒ² + 5x + 6
Expand: (x βˆ’ 3)(x + 7)
⚠️ Middle term trap!
Outer + Inner = 7x + (βˆ’3x) = +4x, not βˆ’4x or βˆ’21!
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8
πŸ“Œ FACTOR: find two numbers that MULTIPLY to c and ADD to b
Factoring Trinomials (xΒ² + bx + c)
✏️ EXAMPLE
Factor: xΒ² + 7x + 10
β†’ Need: Γ— = 10, + = 7 β†’ (2 and 5) β†’ (x+2)(x+5)
Factor completely: xΒ² βˆ’ x βˆ’ 12
⚠️ Signs are tricky here!
Need: multiply = βˆ’12, add = βˆ’1. Try (βˆ’4) and (+3)!
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9
πŸ“Œ SYSTEMS: Substitution β†’ plug one into other
Systems of Equations
✏️ EXAMPLE
y = 2x, x + y = 6 β†’ x + 2x = 6 β†’ 3x = 6 β†’ x = 2, y = 4
Solve the system: y = x + 2
2x + y = 11
⚠️ Don't forget to find BOTH x and y!
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10
πŸ“Œ QUADRATIC FORMULA: x = [βˆ’b Β± √(bΒ²βˆ’4ac)] / 2a
Quadratic Equations
✏️ EXAMPLE
xΒ² βˆ’ 5x + 6 = 0 β†’ factor β†’ (xβˆ’2)(xβˆ’3) = 0 β†’ x = 2 or x = 3
Solve: xΒ² + 2x βˆ’ 8 = 0
⚠️ Two solutions exist!
Factor: find two numbers that multiply to βˆ’8 and add to +2. Try +4 and βˆ’2!
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πŸ“ GEOMETRY β€” Problems 11–20
11
πŸ“Œ TRIANGLE ANGLES = 180Β°, always!
Triangle Angle Sum
✏️ EXAMPLE
Angles 40Β° and 75Β°: 3rd angle = 180Β° βˆ’ 40Β° βˆ’ 75Β° = 65Β°
In β–³ABC, ∠A = 52Β° and ∠B = 3xΒ°, ∠C = (2x βˆ’ 2)Β°. Find x.
⚠️ Don't forget ∠A!
52 + 3x + 2x βˆ’ 2 = 180 β†’ combine β†’ solve!
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12
πŸ“Œ PYTHAGOREAN: aΒ² + bΒ² = cΒ² (c = hypotenuse!)
Pythagorean Theorem
✏️ EXAMPLE
Legs 3 and 4: cΒ² = 9 + 16 = 25 β†’ c = 5
A right triangle has legs of length 5 and 12. What is the hypotenuse?
⚠️ Classic Trap: 5+12β‰ hyp!
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13
πŸ“Œ CIRCLE: Area = Ο€rΒ² Β· Circumference = 2Ο€r
Area & Circumference of Circles
✏️ EXAMPLE
Diameter = 10 β†’ r = 5 β†’ Area = Ο€(5Β²) = 25Ο€
A circle has diameter 14 cm. What is its area? (Use Ο€)
⚠️ diameter β‰  radius!
d = 14, so r = 7! Area = Ο€rΒ² = Ο€(7)Β² = 49Ο€
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14
πŸ“Œ PARALLEL LINES + TRANSVERSAL β†’ alternate angles EQUAL
Parallel Lines & Transversals
✏️ EXAMPLE
Alternate interior angles are equal when lines are parallel.
Co-interior (same-side) angles add up to 180Β°.
Two parallel lines are cut by a transversal. One angle is 110Β°. What is the measure of its co-interior (same-side interior) angle?
⚠️ Co-interior = supplementary!
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15
πŸ“Œ CONGRUENT β‰… same shape & size Β· SAS Β· ASA Β· SSS Β· AAS
Triangle Congruence
✏️ EXAMPLE
If two sides and the included angle of one triangle equal those of another β†’ SAS β†’ Congruent!
Two triangles have: two pairs of equal sides and the included angle equal. Which congruence rule applies?
⚠️ "Included" means between the two sides!
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16
πŸ“Œ MIDPOINT = average of coordinates: ((x₁+xβ‚‚)/2, (y₁+yβ‚‚)/2)
Midpoint Formula
✏️ EXAMPLE
Midpoint of (2, 4) and (6, 8): (2+62, 4+82) = (4, 6)
Find the midpoint of the segment with endpoints (βˆ’3, 7) and (5, βˆ’1).
⚠️ Don't subtract β€” ADD then divide!
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17
πŸ“Œ VOLUME of Prism = Base Area Γ— Height
Volume of 3D Figures
✏️ EXAMPLE
Rectangular prism: l=4, w=3, h=5 β†’ V = 4Γ—3Γ—5 = 60 unitsΒ³
A cylinder has radius 3 cm and height 10 cm. What is its volume? (Use Ο€)
⚠️ V = Ο€rΒ²h β€” don't forget to square r!
V = Ο€ Γ— 3Β² Γ— 10 = Ο€ Γ— 9 Γ— 10 = 90Ο€
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18
πŸ“Œ SIMILAR ~ : ratios of sides EQUAL Β· angles EQUAL
Similar Triangles & Proportions
✏️ EXAMPLE
β–³ABC ~ β–³DEF, AB=6, DE=9, BC=4. Find EF:
ABDE = BCEF β†’ 69 = 4EF β†’ EF = 6
β–³PQR ~ β–³XYZ. PQ = 8, XY = 12, QR = 6. Find YZ.
⚠️ Set up the ratio correctly!
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19
πŸ“Œ EXTERIOR ANGLE of triangle = sum of 2 NON-adjacent interior angles
Exterior Angle Theorem
✏️ EXAMPLE
Interior angles 50Β° and 70Β°, exterior = 50Β° + 70Β° = 120Β°
An exterior angle of a triangle measures 128Β°. Two non-adjacent interior angles are 63Β° and xΒ°. Find x.
⚠️ Don't use 180°!
Exterior = 63 + x β†’ 128 = 63 + x β†’ x = ?
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20
πŸ“Œ DISTANCE = √[(xβ‚‚βˆ’x₁)Β² + (yβ‚‚βˆ’y₁)Β²]
Distance Formula
✏️ EXAMPLE
Distance from (1,1) to (4,5):
√(4βˆ’1)Β² + (5βˆ’1)Β² = √9 + 16 = √25 = 5
Find the distance between (1, 2) and (7, 10).
⚠️ Square BOTH differences!
√[(7βˆ’1)Β² + (10βˆ’2)Β²] = √[36 + 64] = √100 = ?
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