π Algebra 2
Key Topics Β· Common Mistakes Β· Self-Study Edition
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β Wrong: 0
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β οΈ Read the memory point before each problem! It's a lifesaver.
Q1
Quadratic Formula
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π MEMORY POINT
"bΒ² β 4ac" = DISCRIMINANT β decides HOW MANY real solutions
Positive β 2 solutions | Zero β 1 solution | Negative β NO real solution
Positive β 2 solutions | Zero β 1 solution | Negative β NO real solution
βοΈ EXAMPLE
Solve \(x^2 - 5x + 6 = 0\).
\(a=1,\; b=-5,\; c=6\) β \(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\) β \(x=3\) or \(x=2\) β
\(a=1,\; b=-5,\; c=6\) β \(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\) β \(x=3\) or \(x=2\) β
Solve using the quadratic formula: \(2x^2 - 7x + 3 = 0\)
β don't forget to check a=2 !
β don't forget to check a=2 !
π‘ EXPLANATION
\(a=2,\; b=-7,\; c=3\)
Discriminant: \((-7)^2 - 4(2)(3) = 49 - 24 = 25\)
\(x = \dfrac{7 \pm 5}{4}\) β \(x = 3\) or \(x = \dfrac{1}{2}\) β
Common mistake: forgetting to divide by 2a = 4, not 2!
Discriminant: \((-7)^2 - 4(2)(3) = 49 - 24 = 25\)
\(x = \dfrac{7 \pm 5}{4}\) β \(x = 3\) or \(x = \dfrac{1}{2}\) β
Common mistake: forgetting to divide by 2a = 4, not 2!
Q2
Discriminant
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π MEMORY POINT
DISCRIMINANT = bΒ² β 4ac
Just calculate it β don't solve the whole equation!
Just calculate it β don't solve the whole equation!
How many real solutions does \(x^2 + 4x + 5 = 0\) have?
π‘ EXPLANATION
\(b^2 - 4ac = 16 - 20 = -4 < 0\)
Negative discriminant β No real solutions β
Trick: the graph never crosses the x-axis!
Negative discriminant β No real solutions β
Trick: the graph never crosses the x-axis!
Q3
Vertex Form
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π MEMORY POINT
y = a(x β h)Β² + k β Vertex = (h, k)
β οΈ Watch the SIGN! \(y = (x-3)^2\) β vertex is \((+3, 0)\), NOT \((-3,0)\)!
β οΈ Watch the SIGN! \(y = (x-3)^2\) β vertex is \((+3, 0)\), NOT \((-3,0)\)!
βοΈ EXAMPLE
\(y = 2(x-1)^2 + 3\) β vertex \((1, 3)\), opens UP (a=2 > 0)
The parabola \(y = -3(x + 2)^2 - 1\) has vertex at:
π‘ EXPLANATION
\(y = -3(x-(-2))^2 + (-1)\)
So \(h = -2,\; k = -1\) β vertex \((-2, -1)\) β
The sign inside the parenthesis FLIPS! (x+2) means h = β2
So \(h = -2,\; k = -1\) β vertex \((-2, -1)\) β
The sign inside the parenthesis FLIPS! (x+2) means h = β2
Q4
Exponential Growth
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π MEMORY POINT
y = a Β· bΛ£ β Growth if b > 1, Decay if 0 < b < 1
'a' = starting value, 'b' = multiplier each step
'a' = starting value, 'b' = multiplier each step
A bacteria colony starts with 200 cells and doubles every hour.
How many cells after 4 hours?
How many cells after 4 hours?
π‘ EXPLANATION
\(y = 200 \cdot 2^4 = 200 \cdot 16 = 3200\) β
Common mistake: multiplying 200 Γ 4 Γ 2 = 1600 (WRONG! It's exponential, not linear!)
Common mistake: multiplying 200 Γ 4 Γ 2 = 1600 (WRONG! It's exponential, not linear!)
Q5
Logarithms
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π MEMORY POINT
log_b(x) = y β bΚΈ = x
"Log asks: what POWER gives me this number?"
"Log asks: what POWER gives me this number?"
βοΈ EXAMPLE
\(\log_2 8 = ?\) β "2 to what power = 8?" β \(2^3 = 8\) β Answer: 3
Evaluate: \(\log_3 81\)
π‘ EXPLANATION
\(3^? = 81\) β \(3^1=3,\; 3^2=9,\; 3^3=27,\; 3^4=81\) β
So \(\log_3 81 = 4\)
Trick: count how many times you multiply 3 to reach 81
So \(\log_3 81 = 4\)
Trick: count how many times you multiply 3 to reach 81
Q6
Rational Exponents
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π MEMORY POINT
x^(m/n) = (βΏβx)α΅
Denominator = ROOT, Numerator = POWER
Easy trick: "Flower" β bottom = root (like roots of a flower!)
Denominator = ROOT, Numerator = POWER
Easy trick: "Flower" β bottom = root (like roots of a flower!)
Simplify: \(27^{2/3}\)
π‘ EXPLANATION
\(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\) β
Step 1: cube root of 27 = 3 Step 2: square it β 9
Always do the ROOT first β it keeps numbers small and easy!
Step 1: cube root of 27 = 3 Step 2: square it β 9
Always do the ROOT first β it keeps numbers small and easy!
Q7
Systems of Equations
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π MEMORY POINT
SUBSTITUTION: Isolate one variable β plug in β solve β back-substitute
Good for when one equation already says "y = ..."
Good for when one equation already says "y = ..."
Solve the system:
\(y = 2x + 1\)
\(3x + y = 16\)
\(y = 2x + 1\)
\(3x + y = 16\)
π‘ EXPLANATION
Sub \(y = 2x+1\) into \(3x+y=16\):
\(3x + 2x + 1 = 16\) β \(5x = 15\) β \(x=3\)
\(y = 2(3)+1 = 7\) β
Always check: 3(3)+7 = 9+7 = 16 β
\(3x + 2x + 1 = 16\) β \(5x = 15\) β \(x=3\)
\(y = 2(3)+1 = 7\) β
Always check: 3(3)+7 = 9+7 = 16 β
Q8
Polynomial Factoring
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π MEMORY POINT
Difference of Squares: aΒ² β bΒ² = (a+b)(aβb)
No middle term? Check for difference of squares FIRST!
No middle term? Check for difference of squares FIRST!
Factor completely: \(4x^2 - 25\)
π‘ EXPLANATION
\(4x^2 = (2x)^2\) and \(25 = 5^2\)
So \(4x^2 - 25 = (2x+5)(2x-5)\) β
Trap: (2xβ5)Β² would give 4xΒ²β20x+25 β NOT the same! No middle term β not a perfect square.
So \(4x^2 - 25 = (2x+5)(2x-5)\) β
Trap: (2xβ5)Β² would give 4xΒ²β20x+25 β NOT the same! No middle term β not a perfect square.
Q9
Completing the Square
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π MEMORY POINT
Take half of b, then square it: (b/2)Β²
Add AND subtract to keep equation balanced!
Add AND subtract to keep equation balanced!
Complete the square: \(x^2 + 6x + ?\) forms a perfect square trinomial.
What number fills the blank?
What number fills the blank?
π‘ EXPLANATION
\(b = 6\) β \(\left(\dfrac{6}{2}\right)^2 = 3^2 = 9\) β
Check: \(x^2 + 6x + 9 = (x+3)^2\) β
Formula: always use (bΓ·2)Β². Half first, THEN square!
Check: \(x^2 + 6x + 9 = (x+3)^2\) β
Formula: always use (bΓ·2)Β². Half first, THEN square!
Q10
Complex Numbers
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π MEMORY POINT
iΒ² = β1 (ALWAYS remember this!)
Treat 'i' like a variable, but replace iΒ² with β1 at the end!
Treat 'i' like a variable, but replace iΒ² with β1 at the end!
βοΈ EXAMPLE
\((2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 2 + i - 3(-1) = 5 + i\)
Simplify: \((3 + 2i)(3 - 2i)\)
π‘ EXPLANATION
This is a difference of squares: \((a+b)(a-b) = a^2 - b^2\)
\((3)^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13\) β
The imaginary parts always cancel out β real answer!
\((3)^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13\) β
The imaginary parts always cancel out β real answer!
π Geometry
Key Topics Β· Common Mistakes Β· Self-Study Edition
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Correct: 0
β Wrong: 0
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π Draw a quick sketch for each problem β it really helps!
Q1
Triangle Angles
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π MEMORY POINT
Triangle angle sum = 180Β°
All 3 interior angles always add up to EXACTLY 180Β°. No exceptions!
All 3 interior angles always add up to EXACTLY 180Β°. No exceptions!
A triangle has angles measuring \(47Β°\) and \(83Β°\). What is the third angle?
π‘ EXPLANATION
\(180Β° - 47Β° - 83Β° = 50Β°\) β
Quick check: 47 + 83 + 50 = 180 β
Quick check: 47 + 83 + 50 = 180 β
Q2
Pythagorean Theorem
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π MEMORY POINT
aΒ² + bΒ² = cΒ² β c is ALWAYS the longest side (hypotenuse)
Common triples: 3-4-5, 5-12-13, 8-15-17 β MEMORIZE these!
Common triples: 3-4-5, 5-12-13, 8-15-17 β MEMORIZE these!
A right triangle has legs of length \(6\) and \(8\). Find the hypotenuse.
π‘ EXPLANATION
\(6^2 + 8^2 = 36 + 64 = 100\) β \(c = \sqrt{100} = 10\) β
This is the 3-4-5 triple scaled by 2: \((6, 8, 10)\)
Recognize the triple β no calculator needed!
This is the 3-4-5 triple scaled by 2: \((6, 8, 10)\)
Recognize the triple β no calculator needed!
Q3
Parallel Lines & Transversal
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π MEMORY POINT
ALTERNATE INTERIOR angles are EQUAL
CO-INTERIOR (same-side) angles add up to 180Β°
Trick: "Alternate = Equal, Co-interior = 180"
CO-INTERIOR (same-side) angles add up to 180Β°
Trick: "Alternate = Equal, Co-interior = 180"
Two parallel lines are cut by a transversal. One angle is \(65Β°\).
What is the measure of its co-interior angle?
What is the measure of its co-interior angle?
π‘ EXPLANATION
Co-interior angles (same-side interior) are supplementary:
\(180Β° - 65Β° = 115Β°\) β
Co-interior = "C shape" between the parallel lines β adds to 180Β°
\(180Β° - 65Β° = 115Β°\) β
Co-interior = "C shape" between the parallel lines β adds to 180Β°
Q4
Circle: Circumference & Area
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π MEMORY POINT
C = 2Οr | A = ΟrΒ²
Circumference uses r once, Area uses rΒ²
Never mix these up! Area = "r-squared" π₯
Circumference uses r once, Area uses rΒ²
Never mix these up! Area = "r-squared" π₯
A circle has radius \(5\). What is its area? (Leave answer in terms of \(\pi\))
π‘ EXPLANATION
\(A = \pi r^2 = \pi(5)^2 = 25\pi\) β
\(10\pi\) would be the circumference: \(2\pi(5) = 10\pi\)
Area = rΒ², Circumference = 2r β DON'T swap them!
\(10\pi\) would be the circumference: \(2\pi(5) = 10\pi\)
Area = rΒ², Circumference = 2r β DON'T swap them!
Q5
Similar Triangles
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π MEMORY POINT
Similar triangles: corresponding sides are PROPORTIONAL
Set up a ratio and cross-multiply β easy!
Set up a ratio and cross-multiply β easy!
βοΈ EXAMPLE
Triangles with sides 3, 4, 5 and 6, 8, ? are similar β \(\frac{5}{1} \times 2 = 10\)
Two similar triangles. The first has sides \(4, 6, 8\).
The second has shortest side \(6\). Find the longest side of the second triangle.
The second has shortest side \(6\). Find the longest side of the second triangle.
π‘ EXPLANATION
Scale factor: \(\dfrac{6}{4} = 1.5\)
Longest side of second triangle: \(8 \times 1.5 = 12\) β
Always match corresponding sides! Shortest β Shortest, Longest β Longest
Longest side of second triangle: \(8 \times 1.5 = 12\) β
Always match corresponding sides! Shortest β Shortest, Longest β Longest
Q6
Volume of Cylinder
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π MEMORY POINT
V = ΟrΒ²h β Area of circle Γ height
"Stack circles up!" β base area Γ how tall
"Stack circles up!" β base area Γ how tall
A cylinder has radius \(3\) and height \(7\). Find its volume in terms of \(\pi\).
π‘ EXPLANATION
\(V = \pi r^2 h = \pi(3)^2(7) = \pi \cdot 9 \cdot 7 = 63\pi\) β
Common mistake: using diameter instead of radius! Always check if given r or d.
Common mistake: using diameter instead of radius! Always check if given r or d.
Q7
Angle in a Polygon
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π MEMORY POINT
Interior angle sum of n-gon = (n β 2) Γ 180Β°
Hexagon? β (6β2)Γ180 = 720Β°
Think: triangle = 180, quad = 360, pentagon = 540... add 180 each time!
Hexagon? β (6β2)Γ180 = 720Β°
Think: triangle = 180, quad = 360, pentagon = 540... add 180 each time!
What is the sum of the interior angles of a heptagon (7 sides)?
π‘ EXPLANATION
\((7-2) \times 180Β° = 5 \times 180Β° = 900Β°\) β
Count triangles inside the shape: 7-gon = 5 triangles β 5 Γ 180
Count triangles inside the shape: 7-gon = 5 triangles β 5 Γ 180
Q8
Midpoint Formula
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π MEMORY POINT
Midpoint = \(\left(\dfrac{x_1+x_2}{2},\; \dfrac{y_1+y_2}{2}\right)\)
Just AVERAGE the x's and AVERAGE the y's!
Just AVERAGE the x's and AVERAGE the y's!
Find the midpoint of the segment joining \((2, -3)\) and \((8, 7)\).
π‘ EXPLANATION
\(x: \dfrac{2+8}{2} = 5\) \(y: \dfrac{-3+7}{2} = 2\)
Midpoint = \((5, 2)\) β
Don't subtract! Always ADD the coordinates, then divide by 2.
Midpoint = \((5, 2)\) β
Don't subtract! Always ADD the coordinates, then divide by 2.
Q9
30-60-90 Special Triangle
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π MEMORY POINT
30-60-90 sides: 1 : β3 : 2
Shortest (30Β°) β Γ β3 β Medium (60Β°) β Γ 2 β Hypotenuse
45-45-90 sides: 1 : 1 : β2
Shortest (30Β°) β Γ β3 β Medium (60Β°) β Γ 2 β Hypotenuse
45-45-90 sides: 1 : 1 : β2
In a 30-60-90 triangle, the shortest side is \(5\). Find the hypotenuse.
π‘ EXPLANATION
Ratio: short : hypotenuse = 1 : 2
Hypotenuse = \(5 \times 2 = 10\) β
(The middle side would be \(5\sqrt{3}\))
Hypotenuse is ALWAYS double the shortest side in 30-60-90!
Hypotenuse = \(5 \times 2 = 10\) β
(The middle side would be \(5\sqrt{3}\))
Hypotenuse is ALWAYS double the shortest side in 30-60-90!
Q10
Arc Length
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π MEMORY POINT
Arc Length = \(\dfrac{\theta}{360Β°} \times 2\pi r\)
"What fraction of the full circle?" Γ full circumference
Think: 180Β° arc = half circle β Β½ Γ circumference
"What fraction of the full circle?" Γ full circumference
Think: 180Β° arc = half circle β Β½ Γ circumference
βοΈ EXAMPLE
Circle r=6, angle=90Β°: Arc = \(\dfrac{90}{360} \times 2\pi(6) = \dfrac{1}{4} \times 12\pi = 3\pi\)
A circle has radius \(9\) and a central angle of \(120Β°\). Find the arc length in terms of \(\pi\).
π‘ EXPLANATION
\(\dfrac{120}{360} \times 2\pi(9) = \dfrac{1}{3} \times 18\pi = 6\pi\) β
120Β° = β of a circle. So take β of the circumference!
120Β° = β of a circle. So take β of the circumference!