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\fancyhead[LO,LE]{Year 12 Methods}
\fancyhead[CO,CE]{Andrew Lorimer}
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\author{Andrew Lorimer}
\date{}

\begin{document}

\hypertarget{polynomials}{%
\section{Polynomials}\label{polynomials}}

\hypertarget{quadratics}{%
\subsection{Quadratics}\label{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}

\hypertarget{cubics}{%
\subsection{Cubics}\label{cubics}}

\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)

\[y=a(bx-h)^3 + c\]

\begin{itemize}
\tightlist
\item
  \(m=0\) at \emph{stationary point of inflection}
  (i.e.~(\({h \over b}, k)\))
\item
  in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
\item
  in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
\item
  in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
  \(c\)
\end{itemize}

\hypertarget{linear-and-quadratic-graphs}{%
\subsection{Linear and quadratic
graphs}\label{linear-and-quadratic-graphs}}

\hypertarget{forms-of-linear-equations}{%
\subsubsection{Forms of linear
equations}\label{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

\hypertarget{line-properties}{%
\subsection{Line properties}\label{line-properties}}

Parallel lines: \(m_1 = m_2\)\\
Perpendicular lines: \(m_1 \times m_2 = -1\)\\
Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

\hypertarget{quartic-graphs}{%
\subsection{Quartic graphs}\label{quartic-graphs}}

\hypertarget{forms-of-quadratic-equations}{%
\subsubsection{Forms of quadratic
equations}\label{forms-of-quadratic-equations}}

\(y=ax^4\)\\
\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
\(y=ax^4+cd^2 (c \ge 0)\)\\
\(y=ax^2(x-b)(x-c)\)\\
\(y=a(x-b)^2(x-c)^2\)\\
\(y=a(x-b)(x-c)^3\)

\hypertarget{simultaneous-equations-linear}{%
\subsection{Simultaneous equations
(linear)}\label{simultaneous-equations-linear}}

\begin{itemize}
\tightlist
\item
  \textbf{Unique solution} - lines intersect at point
\item
  \textbf{Infinitely many solutions} - lines are equal
\item
  \textbf{No solution} - lines are parallel
\end{itemize}

\hypertarget{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}{%
\subsubsection{\texorpdfstring{Solving
\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
for \(\{0,1,\infty\}\)
solutions}{Solving \textbackslash protect\textbackslash begin\{cases\}px + qy = a \textbackslash\textbackslash{} rx + sy = b\textbackslash protect\textbackslash end\{cases\} \textbackslash\textgreater{} for \textbackslash\{0,1,\textbackslash infty\textbackslash\} solutions}}\label{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}}

where all coefficients are known except for one, and \(a, b\) are known

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Write as matrices:
  \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
\item
  Find determinant of first matrix: \(\Delta = ps-qr\)
\item
  Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
  or let \(\Delta \ne 0\) for one unique solution.
\item
  Solve determinant equation to find variable

  \begin{itemize}
  \tightlist
  \item
    \emph{--- for infinite/no solutions: ---}
  \end{itemize}
\item
  Substitute variable into both original equations
\item
  Rearrange equations so that LHS of each is the same
\item
  \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\)
  (\(\infty\) solns)\\
  \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0
  solns)
\end{enumerate}

\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}

\hypertarget{solving-protectbegincasesa_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_3protectendcases}{%
\subsubsection{\texorpdfstring{Solving
\(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}{Solving \textbackslash protect\textbackslash begin\{cases\}a\_1 x + b\_1 y + c\_1 z = d\_1 \textbackslash\textbackslash{} a\_2 x + b\_2 y + c\_2 z = d\_2 \textbackslash\textbackslash{} a\_3 x + b\_3 y + c\_3 z = d\_3\textbackslash protect\textbackslash end\{cases\}}}\label{solving-protectbegincasesa_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_3protectendcases}}

\begin{itemize}
\tightlist
\item
  Use elimination
\item
  Generate two new equations with only two variables
\item
  Rearrange \& solve
\item
  Substitute one variable into another equation to find another variable
\item
  etc.
\end{itemize}

\end{document}