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\usepackage{booktabs} % Top and bottom rules for tables

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%	TITLE SECTION 
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\title{Quadratic Function} % Poster title

\author{matematika.pl} % Author(s)

\institute{2015} % Institution(s)

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\begin{document}

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\setlength\belowdisplayshortskip{2ex} % White space under equations

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%	OBJECTIVES
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\begin{alertblock}{Objectives}
Objectives for today:
\begin{itemize}
\item Introducing specific vocabulary.
\item Quick revision of quadratic function.
\item Factorising Quadratics.
\item Proving Vieta's formulas.
\item Carrying out gained knowledge by working out some word problems.
\end{itemize}

\end{alertblock}

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%	QUICK REVISION
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\begin{block}{Quick Revision}

\textbf{Forms of Quadratic Function}
\begin{itemize}
\item $f(x) = ax^2+bx+c$ is called the \textbf{standard form}.
\item $f(x) = a(x-x_1)(x-x_2)$ is called the \textbf{factored form}, where $x_1$ and $x_2$ are the roots of the quadratic function.
\item $f(x) = a(x-h)^2+k$ is called the \textbf{vertex form}.
\end{itemize}

\textbf{Delta $\Delta$}\\*
$\Delta$ determines tells us how many solutions quadratic equation have:
$$\text{number of solutions}=
\begin{cases}
2 &\text{when } \Delta > 0\\
1 &\text{when } \Delta = 0\\
0 &\text{when } \Delta < 0
\end{cases}
$$


\textbf{The Quadratic Formula}
$$x = \frac{-b\pm \sqrt{\Delta}}{2a}$$

\textbf{Graph of Quadratic Function}

\end{block}

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% \begin{figure}
% \includegraphics[width=0.8\linewidth]{1.jpg}
% \caption{Graph of $f(x)=ax^2|_{\{0.1, 0.3, 1.0, 3.0\}}$}
% \end{figure}

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%----------------------------------------------------------------------------------------
%	MATERIALS
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\begin{block}{Factorising a Quadratic}

Factorising a quadratic means putting it into two brackets, and is useful if you're trying to draw a graph of a quadratic solve a quadratic equation. It's pretty easy if $a=1$ (in $ax^2+bx+c$ form), but can be a real pain otherwise.
\newline
\newline
In order to factorise a quadratic you should follow steps outlined below:

\begin{enumerate}
\item Rearrange the equation into the standard $ax^2+bx+c$ form.
\item Write down two brackets: $(x\ \ \ )(x\ \ \ )$
\item Find two numbers that multiply to give 'c' and add or subtract to give 'b' (ignoring signs).
\item Put the numbers in brackets and choose their signs.
\end{enumerate}

\end{block}

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%	P
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\begin{block}{Factorising- Tasks}
1. Factorise $x^2-x-12$.
\[\]
\[\]
\[\]
\[\]
2. Solve $x^2-8=2x$ by factorising.

\end{block}

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%----------------------------------------------------------------------------------------
%	IMPORTANT To REMEMBER
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\begin{alertblock}{Myth of Delta $\Delta$}

It's commonly believed that in order to work out roots of a quadratic function you must count $\Delta$ and use other previously established formulas. However this is untrue since factorising in many cases is as good or even better than simply counting $\Delta$.

\end{alertblock} 

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%----------------------------------------------------------------------------------------
%	EXAMPLE OF FACTORISATION
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\begin{block}{Example of Factorisation}
Solve $x^2+4x-21=0$ by factorising.

$$x^2+4x-21=(x\ \ \ \ \ )(x\ \ \ \ \ )$$
$1$ and $21$ multiply to give $21$ - and add or subtract to give $22$ and $20$.\\*
$3$ and $7$ multiply to give $21$ - and add or subtract to give $10$ and \textbf{$4$}.

$$x^2+4x+21 = (x+7)(x-3)$$

And solving the equation:
$$(x+7)(x-3)=0$$
we get
$$x=-7,\ \ \ x=3$$

\end{block}

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%----------------------------------------------------------------------------------------
%	PROOF OF VIETA'S FORMULAS
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\begin{block}{ Proof of Vieta's Formulas}
Let's prove that:
$$x_1 + x_2 = \frac{-b}{a}$$
When $\Delta$ is positive we have two roots:
$$x_1 = \frac{-b-\sqrt{\Delta}}{2a},\ \ \ x_2 = \frac{-b+\sqrt{\Delta}}{2a}$$

Substituting for $x_1$ and $x_2$ respectively, we receive:
					
$$x_1 + x_2 = \frac{-b-\sqrt{\Delta}}{2a} + \frac{-b+\sqrt{\Delta}}{2a} =$$
$$ = \frac{(-b-\sqrt{\Delta}) + (-b+\sqrt{\Delta})}{2a} = \frac{-2b}{2a} = \frac{-b}{a}$$
                    
The same we could do with another pattern, which state that $x_1 x_2 = \frac{c}{a}$, but proving this is going to be your task in next section.

\end{block}

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%	CONCLUSION
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\begin{block}{Vieta's Formulas- Task}
1. Prove that $$x_1x_2 = \frac{c}{a}$$
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\end{block}


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%	ACKNOWLEDGEMENTS
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\setbeamercolor{block title}{fg=red,bg=white} % Change the block title color

\begin{block}{Glossary}

\begin{table}
\vspace{2ex}
\begin{tabular}{l l l l}
\toprule
\textbf{verb} & \textbf{noun} & \textbf{meaning}\\
\midrule
add & addition & $+$ \\
subtract & subtraction & $-$ \\
multiply & multiplication & $\cdot$ \\
divide & division & $\div$ \\
solve & solution & getting answer \\
substitute & substitution & $t=x^2$ \\



\bottomrule
\end{tabular}
\caption{Word Formation}
\end{table}


\end{block}

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\setbeamercolor{block alerted body}{fg=black,bg=white} % Change the alert block body colors

\begin{alertblock}{Some Necessary and Useful Vocabulary}

\begin{itemize}
\item (n.) sign $\rightarrow$ $+$ or $-$
\item (n.) equation $\rightarrow something = 0$ 
\item (n.) factor $\rightarrow$ two multiplied factors give result
\item (v.) factorise $\rightarrow$ putting into brackets
\item (n.) coefficient $\rightarrow$ a constant number i.e. $a$, $b$, $c$ in a pattern $ax^2+bx+c$
\item (n.) quadratic function $\rightarrow$ $f(x) = ax^2+bx+c$
\item (n.) root $\rightarrow$ $\sqrt{sth}$ or solution of quadratic equation
\item (n.) formula $=$ pattern
\end{itemize}

\end{alertblock}


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\end{document}
