Sum of roots = $-\dfrac{b}{a}$
Product of roots = $\dfrac{c}{a}$

$x^2-\text{(sum of roots)}x+\text{(product of roots)}=0$

2 real roots: $b^2-4ac>0$
1 real root (2 equal roots): $b^2-4ac=0$
0 real roots: $b^2-4ac<0$

$b^2-4ac<0$ (because curve has 0 real roots)

Line intersect curve (at 2 points): $b^2-4ac>0$
Line tangent to curve: $b^2-4ac=0$
Line does not intersect curve: $b^2-4ac<0$
*Line meets curve: $b^2-4ac\geq0$

1. $a^m \times a^n = a^{m+n}$
2. $a^m \div a^n = a^{m-n}$
3. $(a^m)^n = a^{mn}$
4. $a^0 = 1$ where $a \neq 0$
5. $a^{-n} = \frac{1}{a^n}$
6. $a^{\frac{1}{n}} = \sqrt[n]{a}$
7. $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$
8. $(a \times b)^n = a^n \times b^n$
9. $(\frac{a}{b})^n = \frac{a^n}{b^n}$

1. $\sqrt{a} \times {\sqrt{a}} = a$
2. $\sqrt{a} \times {\sqrt{b}} = \sqrt{ab}$
3. $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
4. $m \sqrt{a} + n \sqrt{a} = (m+n) \sqrt{a}$
5. $m \sqrt{a} - n \sqrt{a} = (m-n) \sqrt{a}$

For $\dfrac{k}{a\sqrt{b}}$, multiply numerator and denominator by $\sqrt{b}$.
For $\dfrac{k}{a\sqrt{b}+c\sqrt{d}}$, multiply by the conjugate, which is $a\sqrt{b}-c\sqrt{d}$.

$P(x)=\text{divisor}\times Q(x)+R(x)$

If $P(x)$ is divided by $x-c$, remainder is $f(c)$.
If $P(x)$ is divided by $ax-b$, remainder is $f\left(\dfrac{b}{a}\right)$.

If $x+c$ is a factor of $P(x)$, $f(-c)=0$.
If $ax+b$ is a factor of $P(x)$, $f\left(-\dfrac{b}{a}\right)=0$.

$a^3+b^3 = (a+b)(a^2-ab+b^2)$
$a^3-b^3 = (a-b)(a^2+ab+b^2)$

1. $\dfrac{f(x)}{(ax + b)(cx+d)} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d}$

2. $\dfrac{f(x)}{(ax + b)(cx+d)^2} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d} + \dfrac{C}{(cx+d)^2}$

3. $\dfrac{f(x)}{(ax + b)(x^2+c)} = \dfrac{A}{ax+b} + \dfrac{Bx+C}{x^2+c}$

Special case: $\dfrac{f(x)}{(ax + b)(x^2)} = \dfrac{A}{ax+b} + \dfrac{B}{x}+\dfrac{C}{x^2}$

$(a+b)^n = a^n + \binom{n}{1}a^{n-1} b + \binom{n}{2}a^{n-2} b^2 + \ldots + \binom{n}{r}a^{n-r} b^r + \ldots + b^{n}$

$T_{r+1}=\binom{n}{r}a^{n-r} b^r$

$\binom{n}{r}=\dfrac{n!}{r!(n-r)!} = \dfrac{n(n-1)\ldots(n-r+1)}{r!}$

For $|a|=b \Rightarrow a=b$ or $a=-b$

For $\log_a y$ to be defined,
1. $y>0$
2. $a>0, a\ne 1$

1. $\log_{a} x^n = n\log_{a} x$
2. $\log_{a} xy = \log_{a} x + \log_{a} y$
3. $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$
4. $\log_{a} b = \dfrac{\log_{c} b}{\log_{c} a}$
5. $\log_{a} b = \dfrac{1}{\log_{b} a}$

$\log_ay=x \Leftrightarrow y=a^x$
$\lg y=x \Leftrightarrow y=10^x$
$\ln y=x \Leftrightarrow y=e^x$

1. $\text{cosec }\theta=\dfrac{1}{\sin\theta}$

2. $\sec\theta=\dfrac{1}{\cos\theta}$

3. $\cot\theta=\dfrac{1}{\tan\theta}$

1. $\sin(-\theta)=-\sin\theta$
2. $\cos(-\theta)=\cos\theta$
3. $\tan(-\theta)=-\tan\theta$

1. $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

2. $\cot\theta=\dfrac{\cos\theta}{\sin\theta}$

1. $\sin^2{A} + \cos^2{A} = 1$
2. $\sec^2{A}=1+\tan^2{A}$
3. $\text{cosec}^2{A}=1 + \cot^2{A}$

1. $\sin(A \pm B) = \sin A\cos B \pm \cos A \sin B$
2. $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
3. $\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

1. $\sin 2A = 2\sin A \cos A$
2. $\cos2A = \cos^2A - \sin^2A= 2 \cos^2A -1= 1 - 2\sin^2A$
3. $\tan 2A = \dfrac{2\tan A}{1-\tan^2A}$

For $a > 0, b > 0, 0° < \alpha < 90°$,

1. $a\sin \theta \pm b \cos \theta = R \sin (\theta \pm \alpha)$
2. $a \cos \theta \pm b \sin \theta = R \cos (\theta \mp \alpha)$

where $R=\sqrt{a^2+b^2}$, $\tan\alpha=\dfrac{b}{a}$.

1. $-\dfrac{\pi}{2}\leq\sin^{-1}\theta\leq\dfrac{\pi}{2}$

2. $0\leq\cos^{-1}\theta\leq\pi$

3. $-\dfrac{\pi}{2}<\tan^{-1}\theta<\dfrac{\pi}{2}$

1. If $a>0$: Scaling of graph with a factor of $a$ parallel to the $y$-axis
2. If $a<0$: Scaling of graph with a factor of $a$ parallel to the $y$-axis, then reflecting of graph in $x$-axis

For sin & cos: amplitude becomes $|a|$
For tan: there is no amplitude

$\text{amplitude}=\dfrac{\text{maximum} -\text{minimum}}{2}$

Scaling of graph with a factor of $\dfrac{1}{b}$ parallel to the $x$-axis

For sin & cos: period becomes $\dfrac{2\pi}{b}$
For tan: period becomes $\dfrac{\pi}{b}$

Translating of graph by $c$ units parallel to the $y$-axis

$c=\dfrac{\text{maximum}+\text{minimum}}{2}$

1. $y=\sin bx$: Scaling of graph with a factor of $\dfrac{1}{b}$ parallel to the $x$-axis
2. $y=a\sin bx$: Scaling of graph with a factor of $a$ parallel to the $y$-axis (reflecting of graph in $x$-axis if $a<0$)
3. $y=a\sin bx+c$: Translating of graph by $c$ units parallel to the $y$-axis

$m=\dfrac{y_1-y_2}{x_1-x_2}$

$y-y_1=m(x-x_1)$
$y=mx+c$

$\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$

$m_1=m_2$

$m_1=-\dfrac{1}{m_2}$
$m_1\times m_2=-1$

$A=\frac{1}{2}\left| \begin{array}{ccccc} x_1 & x_2 & x_3 & x_4 & x_1\\ y_1& y_2 & y_3 & y_4 & y_1\end{array} \right|$

$=\frac{1}{2}|(x_1y_2+x_2y_3+x_3y_4+x_4y_1)-(x_2y_1+x_3y_2+x_4y_3+x_1y_4)|$

Note: coordinates should be in anti-clockwise direction

$(x-a)^2+(y-b)^2=r^2$
$(a,b)$: centre of circle
$r$: radius

$x^2+y^2+2gx+2fy+c=0$
$(-g,-f)$: centre of circle
$\sqrt{f^2+g^2-c}$: radius

1. $\frac{\text{d}}{\text{d}x} c = 0$
2. $\frac{\text{d}}{\text{d}x} x^n = nx^{n-1}$
3. $\frac{\text{d}}{\text{d}x} \sin x = \cos x$
4. $\frac{\text{d}}{\text{d}x} \cos x = -\sin x$
5. $\frac{\text{d}}{\text{d}x} \tan x = \sec^2 x$
6. $\frac{\text{d}}{\text{d}x} e^{x} = e^{x}$
7. $\frac{\text{d}}{\text{d}x} \ln x = \frac{1}{x}$

Note: $\frac{\text{d}}{\text{d}x} kf(x)=k\times\frac{\text{d}}{\text{d}x} f(x)$

For $y=f(u)$ and $u=g(x)$,
$\dfrac{\text{d}y}{\text{d}x} = \dfrac{\text{d}y}{\text{d}u} \times \dfrac{\text{d}u}{\text{d}x}$

1. $\frac{\text{d}}{\text{d}x} (ax+b)^n = an(ax+b)^{n-1}$
2. $\frac{\text{d}}{\text{d}x} \sin (ax+b) = a\cos (ax+b)$
3. $\frac{\text{d}}{\text{d}x} \cos (ax+b) = -a\sin (ax+b)$
4. $\frac{\text{d}}{\text{d}x} \tan (ax+b) = a\sec^2 (ax+b)$
5. $\frac{\text{d}}{\text{d}x} e^{ax+b} = ae^{ax+b}$
6. $\frac{\text{d}}{\text{d}x} \ln (ax+b) = \frac{a}{ax+b}$

$\dfrac{\text{d}}{\text{d}x}(uv) = u\dfrac{\text{d}v}{\text{d}x} + v\dfrac{\text{d}u}{\text{d}x}$

$\dfrac{\text{d}}{\text{d}x}\left(\dfrac{u}{v}\right)=\dfrac{v\frac{\text{d}u}{\text{d}x}-u\frac{\text{d}v}{\text{d}x}}{v^2}$

1. For increasing functions, $\dfrac{\text{d}y}{\text{d}x}>0$.
2. For decreasing functions, $\dfrac{\text{d}y}{\text{d}x}<0$.

$\dfrac{\text{d}y}{\text{d}t}=\dfrac{\text{d}y}{\text{d}x}\times\dfrac{\text{d}x}{\text{d}t}$

A stationary point is defined when $\dfrac{\text{d}y}{\text{d}x}=0$.

If $\dfrac{\text{d}y}{\text{d}x}=0$ for $x=k$, test for $k^-$, $k$, $k^+$.

Maximum point:

$x$	$k^-$	$k$	$k^+$
$\dfrac{\text{d}y}{\text{d}x}$	$+$	$0$	$-$

Minimum point:

$x$	$k^-$	$k$	$k^+$
$\dfrac{\text{d}y}{\text{d}x}$	$-$	$0$	$+$

Inflexion point:

$x$	$k^-$	$k$	$k^+$
$\dfrac{\text{d}y}{\text{d}x}$	$+$	$0$	$+$
	$-$	$0$	$-$

1. If $\dfrac{\text{d}^2y}{\text{d}x^2}<0$, it is a maximum point.

2. If $\dfrac{\text{d}^2y}{\text{d}x^2}>0$, it is a minimum point.

3. If $\dfrac{\text{d}^2y}{\text{d}x^2}=0$, need to do first derivative test.

1. $\int k\,\text{d}x = kx +c$
2. $\int x^n\,\text{d}x = \dfrac{x^{n+1}}{n+1} +c, n\ne -1$
3. $\int \sin x\,\text{d}x = -\cos x +c$
4. $\int \cos x\,\text{d}x = \sin x +c$
5. $\int \sec^2 x\,\text{d}x = \tan x +c$
6. $\int e^x\,\text{d}x = e^x +c$
7. $\int \frac{1}{x}\,\text{d}x = \ln x +c$

Note: $\int kf(x)\,\text{d}x=k\times\int f(x)\,\text{d}x$

1. $\int (ax+b)^n\,\text{d}x = \dfrac{(ax+b)^{n+1}}{a(n+1)} +c, n \ne -1$

2. $\int \sin (ax+b)\,\text{d}x = -\dfrac{\cos (ax+b)}{a} +c$

3. $\int \cos (ax+b)\,\text{d}x = \dfrac{\sin (ax+b)}{a} +c$

4. $\int \sec^2 (ax+b)\,\text{d}x = \dfrac{\tan (ax+b)}{a} +c$

5. $\int e^{ax+b}\,\text{d}x = \dfrac{e^{ax+b}}{a} +c$

6. $\int \frac{1}{ax+b}\,\text{d}x = \dfrac{\ln (ax+b)}{a}+c$

For $\int f(x) \,\text{d}x = F(x) + c$,
$\displaystyle\int_a^b f(x) \,\text{d}x = F(b) -F(a)$.

For area with respect to $x$-axis, $\displaystyle\int_a^b f(x) \,\text{d}x$.

For area with respect to $y$-axis, $\displaystyle\int_c^d f(y) \,\text{d}y$.

Note: For area below the $x$-axis (taken with respect to $x$-axis) or area to the left of the $y$-axis (taken with respect to $y$-axis), it is taken as negative.

1. $v=\dfrac{\text{d}s}{\text{d}t}$

2. $a=\dfrac{\text{d}v}{\text{d}t}$

3. $s=\displaystyle\int v\,\text{d}t$

4. $v=\displaystyle\int a\,\text{d}t$