Unit 01 β 10
Algebra 2
Core Problems
Core Problems
Quadratics, polynomials, logarithms, complex numbers & sequences. Frequently tested β commonly missed.
01
β‘ MEMORY: "Half-Square"
Take half of b β square it β add both sides
Solve by completing the square:
\[ x^2 - 6x + 5 = 0 \]
\[ x^2 - 6x + 5 = 0 \]
Worked Example
\(x^2 - 4x + 3 = 0\)
β \(x^2 - 4x = -3\)
β \(x^2 - 4x + 4 = 1\)
β \((x-2)^2 = 1\) β \(x = 1\) or \(x = 3\) β
β \(x^2 - 4x = -3\)
β \(x^2 - 4x + 4 = 1\)
β \((x-2)^2 = 1\) β \(x = 1\) or \(x = 3\) β
Write your answers as x = _, _ (smaller first, e.g. "1, 5")
02
β‘ MEMORY: "bΒ²β4ac"
>0 β 2 real roots | =0 β 1 root | <0 β no real roots
For the equation \(2x^2 - 5x + k = 0\) to have exactly one real solution, what must \(k\) equal?
Key Formula
Discriminant \(= b^2 - 4ac\)
Set it equal to 0 for exactly 1 solution.
Set it equal to 0 for exactly 1 solution.
03
β‘ MEMORY: "Plug-In Shortcut"
Remainder when dividing by (xβa) = just plug in x = a
When \(p(x) = x^3 - 3x^2 + 2x - 5\) is divided by \((x - 2)\), what is the remainder?
Remainder Theorem
Remainder = p(2)
Just substitute x = 2 into the polynomial.
Just substitute x = 2 into the polynomial.
04
β‘ MEMORY: "i-cycle"
iΒΉ=i, iΒ²=β1, iΒ³=βi, iβ΄=1 β repeats every 4
Simplify: \( i^{23} \)
Trick
23 Γ· 4 = 5 remainder 3
So \(i^{23} = i^3 = ?\)
So \(i^{23} = i^3 = ?\)
05
β‘ MEMORY: "Product, Quotient, Power"
log(ab)=log a + log b | log(a/b)=log a β log b | log(aβΏ)=nΒ·log a
Solve for \(x\):
\[ \log_2(x+3) + \log_2(x-1) = 5 \]
\[ \log_2(x+3) + \log_2(x-1) = 5 \]
Strategy
Combine using product rule β \(\log_2[(x+3)(x-1)] = 5\)
β \((x+3)(x-1) = 2^5 = 32\)
β Expand, set equal to 0, solve quadratic.
β \((x+3)(x-1) = 2^5 = 32\)
β Expand, set equal to 0, solve quadratic.
06
β‘ MEMORY: "Same Base = Same Exponent"
If bases match β set exponents equal
Solve for \(x\):
\[ 8^{x+1} = 4^{2x-1} \]
\[ 8^{x+1} = 4^{2x-1} \]
Hint
Write both sides as powers of 2:
\(8 = 2^3\), \(4 = 2^2\)
β \(2^{3(x+1)} = 2^{2(2x-1)}\)
\(8 = 2^3\), \(4 = 2^2\)
β \(2^{3(x+1)} = 2^{2(2x-1)}\)
07
β‘ MEMORY: "aβ = aβ + (nβ1)d"
d = common difference | aβ = first term
The 3rd term of an arithmetic sequence is 11, and the 8th term is 31. What is the 20th term?
Two-Step
Step 1: Find d using two given terms.
Step 2: Use aβ = aβ + (nβ1)d to find aββ.
Step 2: Use aβ = aβ + (nβ1)d to find aββ.
08
β‘ MEMORY: "Sβ = aβ(1βrβΏ)/(1βr)"
r = common ratio | works when r β 1
Find the sum of the first 5 terms of a geometric sequence where \(a_1 = 3\) and \(r = 2\).
09
β‘ MEMORY: "Denominator β 0"
Always exclude values that make the bottom zero
Simplify and state the domain restriction:
\[ \frac{x^2 - 9}{x^2 - x - 6} \]
\[ \frac{x^2 - 9}{x^2 - x - 6} \]
Factor both
Numerator: \(x^2-9 = (x+3)(x-3)\)
Denominator: \(x^2-x-6 = (x-3)(x+2)\)
Denominator: \(x^2-x-6 = (x-3)(x+2)\)
Write simplified form as a fraction, e.g. "(x+3)/(x+2)"
10
β‘ MEMORY: "Swap x and y, then solve"
Replace f(x) with y β swap xβy β solve for y
Find \(f^{-1}(x)\) if \(f(x) = \dfrac{2x+1}{x-3}\).
Method
y = (2x+1)/(xβ3)
Swap: x = (2y+1)/(yβ3)
β Solve for y β that's fβ»ΒΉ(x)
Swap: x = (2y+1)/(yβ3)
β Solve for y β that's fβ»ΒΉ(x)
Write as "(3x+1)/(x-2)" format
Unit 11 β 20
Geometry 2
Core Problems
Core Problems
Circles, similar triangles, trigonometry, coordinate geometry & proofs. Focus on commonly missed concepts.
11
β‘ MEMORY: "Inscribed = Half Central"
Inscribed angle = Β½ Γ arc it intercepts
A central angle intercepts an arc of 140Β°. What is the inscribed angle that intercepts the same arc?
Rule
Inscribed angle = Β½ Γ (intercepted arc)
12
β‘ MEMORY: "AA β Similar β Proportional"
2 equal angles β similar β set up proportion
Triangles ABC and DEF are similar. \(AB = 6\), \(BC = 9\), \(DE = 10\). Find \(EF\).
Proportion Setup
\(\dfrac{AB}{DE} = \dfrac{BC}{EF}\)
\(\dfrac{6}{10} = \dfrac{9}{EF}\)
\(\dfrac{6}{10} = \dfrac{9}{EF}\)
13
β‘ MEMORY: "a/sinA = b/sinB = c/sinC"
Side over opposite angle β all equal
In triangle ABC: \(\angle A = 30Β°\), \(\angle B = 45Β°\), \(a = 8\). Find side \(b\) (round to 2 decimal places).
Setup
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
\(\dfrac{8}{\sin 30Β°} = \dfrac{b}{\sin 45Β°}\)
\(\dfrac{8}{\sin 30Β°} = \dfrac{b}{\sin 45Β°}\)
14
β‘ MEMORY: "Midpoint = Average, Distance = Pythagoras"
M = ((xβ+xβ)/2, (yβ+yβ)/2)
Points \(A(2, -3)\) and \(B(8, 5)\) are endpoints of a diameter of a circle. What is the area of the circle? (Use \(\pi\), exact answer.)
Two-Step
Step 1: Find diameter using distance formula.
Step 2: radius = diameter / 2 β Area = ΟrΒ²
Step 2: radius = diameter / 2 β Area = ΟrΒ²
Write as "25Ο" format
15
β‘ MEMORY: "Tangent-Chord = Half Arc"
Angle between tangent and chord = Β½ Γ intercepted arc
A tangent and a chord meet at point P on a circle. The intercepted arc is 110Β°. Find the angle between the tangent and the chord.
16
β‘ MEMORY: "Apply Twice"
3D diagonal: d = β(lΒ²+wΒ²+hΒ²)
A rectangular box has dimensions \(3 \times 4 \times 12\). Find the length of the space diagonal (corner to corner).
Formula
\(d = \sqrt{l^2 + w^2 + h^2}\)
17
β‘ MEMORY: "30-60-90: 1 β β3 β 2"
Shortest: x | Middle: xβ3 | Hypotenuse: 2x
In a 30-60-90 triangle, the hypotenuse is 16. Find the shorter leg and the longer leg.
Ratio
Short : Long : Hyp = 1 : β3 : 2
Hypotenuse = 16, so short = 8, long = ?
Hypotenuse = 16, so short = 8, long = ?
Answer as "8, 8β3"
18
β‘ MEMORY: "Outside Γ Whole = Outside Γ Whole"
external segment Γ whole secant = same on other side
Two secants from external point P. One secant: external = 4, whole = 10. Other secant: external = 5, whole = \(x\). Find \(x\).
Formula
\(4 \times 10 = 5 \times x\)
19
β‘ MEMORY: "Sector = (ΞΈ/360) Γ ΟrΒ²"
Just a fraction of the full circle area
A circle has radius 9 cm and a central angle of 80Β°. Find the area of the sector. (Leave answer in terms of \(\pi\).)
Write as "18Ο" format
20
β‘ MEMORY: "Parallel = Same Slope"
mβ = mβ β parallel | mβ Γ mβ = β1 β perpendicular
Line 1 passes through \((1, 2)\) and \((4, 8)\). Line 2 passes through \((0, -1)\) and \((3, 5)\). Are these lines parallel, perpendicular, or neither?
Slope Formula
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)