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 4. More about Polynomials (Calculator Program to check answer) ➢ Find roots ↔ Factorization 5. More about Equations { 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