+---
+geometry: a4paper, margin=2cm
+columns: 2
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+- \usepackage[dvipsnames]{xcolor}
+---
+
+\setstretch{1.3}
+\definecolor{cas}{HTML}{e6f0fe}
+\pagenumbering{gobble}
+\renewcommand{\arraystretch}{1.4}
+
# Polynomials
-## Factorising
+## Quadratics
+
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\begin{tabularx}{\columnwidth}{Rl}
+ General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
+ \hline
+ Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
+ \hline
+ Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
+ \hline
+ Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+ \hline
+ Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
+\end{tabularx}
-#### Quadratics
-**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$
-**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$
-**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$
-**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
-**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$
-**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
+## Cubics
-#### Cubics
**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
+$$y=a(bx-h)^3 + c$$
+
+- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
+- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
+- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
+- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
+
## Linear and quadratic graphs
### Forms of linear equations
Parallel lines: $m_1 = m_2$
Perpendicular lines: $m_1 \times m_2 = -1$
-Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
-
-
-## Cubic graphs
-
-$$y=a(x-b)^3 + c$$
-
-- $m=0$ at *stationary point of inflection*
-- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
-- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
-
+Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
## Quartic graphs
$y=a(x-b)^2(x-c)^2$
$y=a(x-b)(x-c)^3$
-## Literal equations
-
-Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
-
## Simultaneous equations (linear)
- **Unique solution** - lines intersect at point
- **Infinitely many solutions** - lines are equal
- **No solution** - lines are parallel
-### Solving $\begin{cases}px + qy = a \\ rx + sy = b\end{cases}$ for one, infinite and no solutions
+### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>$ for $\{0,1,\infty\}$ solutions
where all coefficients are known except for one, and $a, b$ are known
- *--- for infinite/no solutions: ---*
5. Substitute variable into both original equations
6. Rearrange equations so that LHS of each is the same
-7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)
- If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
+7. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns)
+ $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns)
-Or use Matrix -> `det` on CAS.
+\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det`
-### Solving $\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
+### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
a_2 x + b_2 y + c_2 z = d_2 \\
-a_3 x + b_3 y + c_3 z = d_3\end{cases}$
+a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
- Use elimination
- Generate two new equations with only two variables
- Rearrange & solve
- Substitute one variable into another equation to find another variable
-- etc.
\ No newline at end of file
+- etc.