+---
+geometry: margin=1.5cm
+columns: 2
+header-includes:
+- \usepackage{tabularx}
+---
+
+\pagenumbering{gobble}
+\renewcommand{\arraystretch}{1.4}
+
# Polynomials
-## Factorising
+## Quadratics
-#### 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)
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\begin{tabularx}{\columnwidth}{|R|l|}
+ 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 & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+ Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
+\end{tabularx}
+
+## 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:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
## Linear and quadratic graphs
-$$y=mx+c, \quad {x \over a} + {y \over b}=1$$
+### Forms of linear equations
+
+$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept
+${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph
+$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
-Parallel lines - $m_1 = m_2$
-Perpendicular lines - $m_1 \times m_2 = -1$
+## Line properties
+
+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
-$$y=a(x-b)^3 + c$$
+$$y=a(bx-h)^3 + c$$
-- $m=0$ at *stationary point of inflection*
+- $m=0$ 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 $c$
## Quartic graphs
-$$y=ax^4$$
-
-$$=a(x-b)(x-c)(x-d)(x-e)$$
-
-$$=ax^4+cd^2 (c \ge 0)$$
-
-$$=ax^2(x-b)(x-c)$$
-
-$$=a(x-b)^2(x-c)^2$$
-
-$$=a(x-b)(x-c)^3$$
-
-where
-- $x$-intercepts at $x=b,c,d,e$
+### Forms of quadratic equations
+$y=ax^4$
+$y=a(x-b)(x-c)(x-d)(x-e)$
+$y=ax^4+cd^2 (c \ge 0)$
+$y=ax^2(x-b)(x-c)$
+$y=a(x-b)^2(x-c)^2$
+$y=a(x-b)(x-c)^3$
## Literal equations
-Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters))
+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 in matrix form - use inverse $A^{-1}= {1 \over {ad-bc}}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$. $A^{-1}$ exists for infinite solutions or no solutions ($ad-bc=0$), does not exist for unique solutions ($ad-bc \ne 0$).
-Or use `det` on CAS.
\ No newline at end of file
+- **Unique solution** - lines intersect at point
+- **Infinitely many solutions** - lines are equal
+- **No solution** - lines are parallel
+
+### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
+
+where all coefficients are known except for one, and $a, b$ are known
+
+1. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
+ \begin{bmatrix} x \\ y \end{bmatrix}
+ =
+ \begin{bmatrix} a \\ b \end{bmatrix}$
+2. Find determinant of first matrix: $\Delta = ps-qr$
+3. Let $\Delta = 0$ for number of solutions $\ne 1$
+ or let $\Delta \ne 0$ for one unique solution.
+4. Solve determinant equation to find variable
+ - *--- 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)
+
+Or use Matrix -> `det` on CAS.
+
+### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
+a_2 x + b_2 y + c_2 z = d_2 \\
+a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
+
+- Use elimination
+- Generate two new equations with only two variables
+- Rearrange & solve
+- Substitute one variable into another equation to find another variable
+- etc.