# Extra ```1. Common Identities about Sum and Product of Roots of Quadratic Equation
(a)
α 2+ β 2=( α +β )2−2 α β
(b) α 3+ β 3=( α + β )(α 2−α β + β 2 )=( α + β )[( α + β )2−3 α β ]
(c)
α 3−β 3 =( α −β )( α 2+α β + β 2 )=( α −β )[( α +β )2−α β ]
2
(d) (α − β )2=(α +β )2−4 α β → α −β =±√(α + β ) −4 α β
2. Complex no. (Use of Calculator)
3. Graphs of Quadratic Function y=ax 2+bx+c
➢ b : coefficient of x → slope at y-intercept
y
b>0
y
x
x
y
y
b<0
x
x
➢ Find roots ↔ Factorization
{
y=a x 2+b x+c → mx+d =a x 2+b x+c → 0=a x 2+( b−m) x+(c−d )
y=mx+d
y
➢ Roots of this equation = x-coordinate of the intersection point(s).
➢ Mid. pt. x =
−(b−m)
(Remember, sum of roots / 2)
2a
➢ Mid. pt. y → Sub x into STRAIGHT LINE.
➢ Distance between them =
√1+m2 (α −β )
x
6. Variation
➢ Find constant expression → k as subject → +−×÷ / reciprocal / square / root / log ...
yz
x
k x
e.g. z ∝ √ → z= √ → k =
y
y
√x
i.e.
yz
or
√x
√ x or y 2 z 2 or √ yz or ...
4
yz
x
√x
7. Arithmatic and Geometric Sequence
(a) If Tn is a A.S.
➢
A×T n +B is a A.S.
➢
C
Tn
is a G.S.
(b) If Tn is a G.S.
➢
A×T n is a G.S.
➢
(T n )C is a G.S.
➢
log (T n) is a A.S. , if Tn is all positive.
(c) Multiplication of G.S.
➢ e.g. T (1)×T (2)×T (3)×...×T (10)=ar 0×ar 1×ar 2×...×ar 9=a9 r 0+1+2+...+9
➢ A.S. Sum in the index of r.
8. Equation of Straight Line
Given x-intercept = a and y-intercept = b →
x y
+ =1
a b
9. Equation of Circle ( x−h)2+( y−k )2=r 2 / x 2+ y 2+Dx+Ey+F =0
(a) Point (inside / on / outside) the circle
➢ Sub point into L.H.S.
➢ L.H.S. > R.H.S. → Outside the circle
➢ L.H.S. = R.H.S. → On the circle
➢ L.H.S. < R.H.S. → Inside the circle
(b)
{
x 2 + y 2+Dx+Ey+F =0 (Calculator program)
Ax+ By+C=0
10. Permutations and Combinations
(a) Assign people to distinct groups
➢ e.g. 12 students, 4 different project → 12C3×9C3×6C3×3C3
➢ e.g. 12 students, 5, 4, 3 students as a group → 12C5×7C4×3C3
(b) Assign people to indistinct groups
➢ e.g. 12 students, 4 groups with same project → 12C3×9C3×6C3×3C3÷4 !
➢ e.g. 12 students, 4, 4, 4 students as a group → 12C4×8C4×4C4÷3 !
➢ e.g. 12 students, 4, 3, 3, 2 students as a group → 12C4×8C3×5C3×2C2÷2 !
➢ e.g. 12 students, 4, 4, 2, 2 students as a group → 12C4×8C4×4C2×2C2÷2 !÷2!
11. Measures of Dispersion
(a) 25% data increases / decreases
➢ Compare:
Min. Q1
Med. Q3
Max.
Med. Q3
Max.
(b) 50% data increases / decreases
➢ Compare:
Min. Q1
(c) Adding a datum with value = mean
➢ mean, range, IQR : unchange
➢ S.D. , variance : decrease OR UNCHANGE (Special case: all datum are the same)
12. Trigonometry
c
(a) 3D Pyth. Thm
➢
a 2+b2+c 2=d 2
d
b
a
13. Regular Tetrahedron
(a) Side Length (a) and Height (h)
➢
√
2
h= a
3
14. Common Similar Triangles
a
h
```