科學計算機及數學程式 Scientific Calculator and Mathematical Programs

EXE 科學計算機 Scientific Calculator 具分數顯示功能 Support fraction display

(x,y) 聯立方程求解器 Simultaneous Equation Solver 支援分數及小數輸入 Support fraction and decimal number

ax²+bx+c=0 二次方程求解器 Quadratic Equation Solver 不含因式分解功能 Without Factorization

ax³+bx²+cx+d=0 三次方程求解器 Cubic Equation Solver 因式分解功能 Factorization

(x+y)(x-y)=(x)²-(y)² 多項式展開 Polynomial Expansion e.g. (2)(x+1),(x+1)(x+1),(x+1)(x+2),etc

ax²+bx+c 二次多項式 Quadratic Polynomial 具因式分解功能 Factorization

√ 根式化簡器 Surd simplifier √(a²)=a,√(ab)=√a√b,√(a²b)=a√b

5** Important Formulas

Quadratic Equations in One Unknown

For the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+c=0$, where $a\ne 0$:

(a) $x=\frac{-b\pm \sqrt{b^2-4\mathrm{ac}}}{2a}$

(b) Discriminant $\Delta =b^2-4\mathrm{ac}$

• $\Delta >0$ for the equation has 2 roots
• $\Delta =0$ for the equation has only 1 root (or 2 equal roots)
• $\Delta <0$ for the equation has no roots
• $\Delta \ge 0$ for the equation has root(s) (can be 1 or 2)

(c)

• (i) Sum of roots = $\alpha +\beta =-\frac{b}{a}$
• (ii) Product of roots = $\mathrm{\alpha \beta }=\frac{c}{a}$

Note: ${\alpha }^2+{\beta }^2=(\alpha +\beta ){}^2-2\mathrm{\alpha \beta }$

New equation: $x^2-(\mathrm{\alpha \text{'}}+\mathrm{\beta \text{'}})x+\mathrm{\alpha \text{'}\beta \text{'}}=0$

Quadratic Graph

• $a>0$: open upwards, minimum
• $a<0$: open downwards, maximum
• $c$ = y-intercept = Sub $(0,y)$, that is $x=0$, find $y$

Vertex = $(-\frac{b}{2a},-\frac{b^2-4\mathrm{ac}}{4a})=(h,k)$

If $y=f\left(x\right)=a(x-h){}^2+k$:

• Axis of symmetry: $x=-\frac{b}{2a}$ [or $x=h$]
• Optimum/Maximum/Minimum = $-\frac{b^2-4\mathrm{ac}}{4a}$ [or $k$]

Other:

1. $x$-intercept = Sub $(x,0)$, that is $y=0$, find $x$
2. Completing square: $y=a{x}^2+\mathrm{bx}+c=a(x+\frac{b}{2a}){}^2+c-\frac{b^2}{4a}$
3. Important Skill: Write down all coordinates. Sub those points lying on the graph into the equations one by one.

Equation of Straight Lines

Find slope first.

1. (a) For the line passing through a point $(x_1,y_1)$ with slope $m$:
   equation of the straight line: $y-y_1=m(x-x_1)$
2. (b) $y=\mathrm{mx}+c$, $m$ is the slope, $c$ is the y-intercept
3. (c) For an equation of the straight line $\mathrm{Ax}+\mathrm{By}+C=0$:
   • slope = $-\frac{A}{B}$
   • y-intercept = $-\frac{C}{B}$
   • x-intercept = $-\frac{C}{A}$

Find intersection point: Solve equations $L_1$ and $L_2$.

More about Polynomials

1. (a) Division algorithm: Dividend = Divisor × Quotient + Remainder
   e.g., $7=2\times 3+1$
   Let remainder be $R$ [constant] or $\mathrm{Ax}+B$ [linear] or $A{x}^2+\mathrm{Bx}+C$ [Quadratic]
2. (b) A polynomial $P\left(x\right)$ is divided by $\mathrm{ax}-b$, the remainder $R$ is $P\left(\frac{b}{a}\right)$
3. (c) Long Division

Exponential and Logarithmic Functions

1. (a) $a^{\frac{m}{n}}=\sqrt[n]{a^m}$, where $m$ and $n$ are integers and $n>0$
2. (b) Laws of indices:
   1. (i) $a^m\times a^n=a^{m+n}$
   2. (ii) $a^m÷a^n=a^{m-n}$
   3. (iii) $\left(a^m\right){}^n=a^{\mathrm{mn}}$
   4. (iv) $\left(\mathrm{ab}\right){}^m=a^m{b}^m$
   5. (v) $\left(\frac{a}{b}\right){}^m=\frac{a^m}{b^m}$
   5 methods to solve:
   1. $logx=logA$ → $x=A$
   2. $A^x=B$ → $xlogA=logB$
   3. ${log}_{A}x=B$ → $x=A^B$
   4. $A^{x+1}+A^x=B$ → Factorize: $A^x(A+1)=B$
   5. $A^{2x+1}+A^x=B$ → Let $y=A^x$: $A{y}^2+y=B$
3. (c) For $M,N>0$, $a>0$ and $a\ne 1$:
   1. (i) ${log}_a\left(\mathrm{MN}\right)={log}_{a}M+{log}_{a}N$
   2. (ii) ${log}_a\left(\frac{M}{N}\right)={log}_{a}M-{log}_{a}N$
   3. (iii) ${log}_a\left(M^n\right)=n{log}_{a}M$
   4. (iv) ${log}_{a}1=0$
   5. (v) ${log}_{a}a=1$
   [Change Base] ${log}_{a}b=\frac{{log}_{c}b}{{log}_{c}a}=\frac{logb}{loga}$
4. (d) $L={log}_{10}\left(\frac{I}{I_0}\right)$, where $I$ is the intensity of sound and $I_0$ is the minimum audible sound intensity
5. (e) $logE=4.8+1.5R$, where $E$ is the energy released (in joules (J)) from an earthquake

More about Trigonometry

1. (a) Trigonometric identities:
   1. (i) $90°$, $270°$ changed: sin ↔ cos, cos ↔ sin, tan ↔ 1/tan
   2. (ii) $0°$, $180°$, $360°$ unchanged: sin ↔ sin, cos ↔ cos, tan ↔ tan
2. (b)
   1. (i) area of $△\mathrm{ABC}=\frac{1}{2}\mathrm{ab}sinC$
   2. (ii) Heron's Formula: area of $△\mathrm{ABC}=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{1}{2}(a+b+c)$
   3. (iii) Sine Formula: $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$
   4. (iv) Cosine Formula: $c^2=a^2+b^2-2\mathrm{ab}cosC$ or $cosC=\frac{a^2+b^2-c^2}{2\mathrm{ab}}$
   $a$, $b$ and $c$ are the opposite sides of $\angle A$, $\angle B$ and $\angle C$ respectively.

Variation

• Directly: $k\times$
• Inversely: $k÷$
• Partly: $k_1$, $k_2$ constant
• square: $\left(\right){}^2$
• square root: $\sqrt{}$
• cube: $\left(\right){}^3$
• cubic root: $\sqrt[3]{}$

Permutation and Combination

1. (a) The number of permutations of $r$ objects from $n$ distinct objects without repetition = ${}_{n}P_r=\frac{n!}{(n-r)!}$ [Consider Order]
2. (b) The number of combinations of $r$ objects from $n$ distinct objects without repetition = ${}_{n}C_r=\frac{{}_{n}P_r}{r!}=\frac{n!}{r!(n-r)!}$ [Not Consider Order]

More about Probability

1. (a) If two events $E$ and $F$ are mutually exclusive, then $P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)$
2. (b) For any two events $E$ and $F$, $P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right)$
3. (c) The complementary event of $E$ is denoted by $E\text{'}$. $P(E\text{'})=1-P\left(E\right)$
4. (d) If two events $E$ and $F$ are independent, then $P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\right)$
5. (e) Conditional probability: For any two events $E$ and $F$, $P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\text{after}E\text{has occurred}\right)$, where $P\left(E\right)>0$

More about Statistics

1. (a) Range Range = Largest value − Smallest value Range = Highest class boundary − Lowest class boundary
2. (b) Inter-quartile range Inter-quartile range = Upper quartile − Lower quartile
3. (c) Standard deviation $\sigma =\sqrt{\frac{{(x_1-\overline{x})}^2+{(x_2-\overline{x})}^2+\backslash cdots+{(x_n-\overline{x})}^2}{n}}$ where x ¯ is the mean and n is the total number of data.
4. (d) Standard score = $\frac{x-\overline{x}}{\sigma }$ , where x is the value of the datum.
5. Normal distribution: 34%, 13.5%, 2.35%, 0.15%
6. If each datum is added/subtracted by k: mean, mode, median → add/subtract k Range, IQR, S.D., Variance → no change
7. If each datum is multiplied/divided by k: mean, mode, median → multiply/divide by k Range, IQR, S.D. → multiply/divide by k Variance → multiply/divide by k squared

Equations of Circles

1. For a circle with the centre (h, k) and radius r, the equation of the circle: (x - h)^2 + (y - k)^2 = r^2
2. For an equation of a circle x^2 + y^2 + Dx + Ey + F = 0, centre = ( -D/2, -E/2 ) radius = sqrt{ (D/2)^2 + (E/2)^2 - F }
3. Real Circle: r > 0 Point Circle: r = 0 Imaginary Circle: r < 0
4. *Equation of Tangent: Let y = mx + c. Sub y = mx + c into x^2 + y^2 + Dx + Ey + F = 0, find Δ = b^2 − 4ac.

5.                         *Geometric Relationships:
                           Collinear: A, B, C forms a straight line.
                           Concyclic: A, B, C, D forms a circle.
                           Perpendicular bisector: straight line passes through mid-point of AB and
                               perpendicular to
                               AB.
                           Angle bisector: two straight lines pass through intersection of AB and CD, and
                               divide
                               angles equally.
                           Parallel: slope of straight line = slope of AB.
                           Perpendicular: (slope of line) × (slope of AB) = -1.
                           

Arithmetic and Geometric Sequences

1. Arithmetic sequence (AS):
2. a = first term, d = common difference, n = number of terms, l = last term
3. General term: T(n) = a + (n - 1)d
4. Sum: S(n) = n/2 [2a + (n - 1)d] or S(n) = n/2 (a + l)
5. Geometric sequence (GS):
6. a = first term, r = common ratio, n = no. of terms
7. General term: T(n) = a r^{n-1}
8. Sum: S(n) = [a(r^n - 1)]/(r - 1) or S(n) = [a(1 - r^n)]/(1 - r)
9. If $-1\mathrm{<\; r\; <}1$, sum to infinity: S(∞)=a/(1 - r)

Compound Inequality

1. “And”: overlap， "Or": All
2. a is positive：
3. ax^2 + bx + c < 0， roots α < x < β
4. ax^2 + bx + c ≤ 0， α ≤ x ≤ β
5. ax^2 + bx + c > 0, x < α or x> β
6. ax^2 + bx + c ≥ 0， x ≤ α or x ≥ β

Percentage change = (New value − Original value) / Original value × 100% New value = Original value × (1 + Percentage increase) New value = Original value × (1 − Percentage decrease) Profit and loss Percentage = (Selling price − Cost price) / Cost price × 100% If the percentage change > 0, there is a profit; if < 0, there is a loss. Selling price = Cost price × (1 + Profit percentage) Selling price = Cost price × (1 − Loss percentage) Discount percentage = (Marked price − Selling price) / Marked price × 100% Selling price = Marked price × (1 − Discount percentage) Square: Perimeter = 4 × Side length; Area = (Side length)^2 Rectangle: Perimeter = 2 × (Length + Width); Area = Length × Width Triangle: Perimeter = sum of three sides; Area = (1/2) × Base × Height Parallelogram: Perimeter = 2 × (sum of 2 sides); Area = Base × Height Trapezium: Perimeter = sum of 4 sides; Area = (1/2) × (sum of two bases) × Height Rhombus: Perimeter = 4 × Side length; Area = Base × Height Cube: Surface area = 6 × (Side length)^2; Volume = (Side length)^3 Cuboid: Surface area = 2(Length × Width + Width × Height + Height × Length); Volume = Length × Width × Height Prism: Surface area = All lateral faces + 2 × Base area; Volume = Base area × Height Rotation 90°: (x, y) → (±y, ±x) Rectangular to polar: r = sqrt(x^2 + y^2), θ = arctan(y/x) Polar to rectangular: x = r cos θ, y = r sin θ Pie Chart: θ° / 360° = ?% / 100% = Part / Total