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<p><strong>Vieta's</strong> Formulae relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète. </p>
<p>For a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ with roots ${{x}_{1}}$ and ${{x}_{2}}$, it is satisfied that </p>
<p style="text-align: center">Sum of Roots: ${{x}_{1}}+{{x}_{2}}=-\frac{b}{a}$</p>
<p style="text-align: center">Product of Roots:  ${{x}_{1}}{{x}_{2}}=\frac{c}{a}$</p>
<p>Show that for a cubic equation in the form of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ with roots ${{x}_{1}}$, ${{x}_{2}}$ and ${{x}_{3}}$, Vieta’s Formulae are given by</p>
<p style="text-align: center">$\begin{matrix}
  {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-\frac{b}{a} \\ 
  {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{2}}{{x}_{3}}=\frac{c}{a} \\ 
  {{x}_{1}}{{x}_{2}}{{x}_{3}}=-\frac{\mathrm{d}}{a} \\ 
\end{matrix}$</p>

**Vieta's** Formulae relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète. 

For a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ with roots ${{x}_{1}}$ and ${{x}_{2}}$, it is satisfied that 

Sum of Roots: ${{x}_{1}}+{{x}_{2}}=-\frac{b}{a}$

Product of Roots:  ${{x}_{1}}{{x}_{2}}=\frac{c}{a}$

Show that for a cubic equation in the form of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ with roots ${{x}_{1}}$, ${{x}_{2}}$ and ${{x}_{3}}$, Vieta’s Formulae are given by

$\begin{matrix}
  {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-\frac{b}{a} \\ 
  {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{2}}{{x}_{3}}=\frac{c}{a} \\ 
  {{x}_{1}}{{x}_{2}}{{x}_{3}}=-\frac{\mathrm{d}}{a} \\ 
\end{matrix}$

IP3U2 HOT PN1 Q8

Modulus Function

<ol class="abc-list"><li><div class="list-item-paragraph">If $x$ and $y$ are positive integers and $x\ne y$, prove that $xy\left( x+y \right)&lt;{{x}^{3}}+{{y}^{3}}$.</div></li><li><div class="list-item-paragraph">If $a$, $b$, $c$ and $d$ are real numbers such that $a+b+c+d=1$, prove that ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\ge \frac{1}{4}$.<br><br></div></li></ol>

IP3U2 HOT PN1 Q7

<p>Find the exact value of $1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots }}}$.<br></p>

Find the exact value of $1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots }}}$.


IP3U2 HOT PN1 Q6

<p>Find the exact value of $\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$.</p>

Find the exact value of $\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$.

IP3U2 HOT PN1 Q5

<p>Solve the inequality $\frac{2x-{{x}^{2}}-5}{(x+1)(x-3)}\le 0$. Hence, solve the following inequalities:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{2x-{{(x+2)}^{2}}-1}{(x+3)(x-1)}\le 0$</div></li><li><div class="list-item-paragraph">$\frac{2\left| x \right|-{{x}^{2}}-5}{(\left| x \right|+1)(\left| x \right|-3)}\le 0$</div></li><li><div class="list-item-paragraph">$\frac{2{{x}^{2}}-{{x}^{4}}-5}{({{x}^{2}}+1)({{x}^{2}}-3)}\le 0$</div></li></ol>

Solve the inequality $\frac{2x-{{x}^{2}}-5}{(x+1)(x-3)}\le 0$. Hence, solve the following inequalities:

IP3U2 HOT PN1 Q4

<p>Solve each of the following inequalities.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{{{\alpha }^{2}}+2}{\alpha -1}&lt;0$</div></li><li><div class="list-item-paragraph">$\frac{{{\alpha }^{2}}-4}{\alpha }&gt;0$</div></li><li><div class="list-item-paragraph">$\left( x+4 \right)\left( 2x-1 \right)\left( x-3 \right)\ge 0$</div></li><li><div class="list-item-paragraph">$\left( 2x+3 \right)\left( x-1 \right)\left( 2-x \right)&lt;0$</div></li><li><div class="list-item-paragraph">$\frac{\left( 3x+4 \right)\left( x-5 \right)}{2x-1}\ge 0$</div></li></ol>

Solve each of the following inequalities.

IP3U2 HOT PN1 Q3

<p>Solve $3{{\left( x-\frac{1}{x} \right)}^{2}}-13x+\frac{13}{x}-10=0$, leaving your answers in exact form.</p>

Solve $3{{\left( x-\frac{1}{x} \right)}^{2}}-13x+\frac{13}{x}-10=0$, leaving your answers in exact form.

IP3U2 HOT PN1 Q2

<p>Solve $\frac{1}{x-6}-\frac{1}{x-4}=\frac{1}{x-3}-\frac{1}{x-5}$, leaving your answers in exact form.</p>

Solve $\frac{1}{x-6}-\frac{1}{x-4}=\frac{1}{x-3}-\frac{1}{x-5}$, leaving your answers in exact form.

IP3U2 HOT PN1 Q1

<p>Solve each of the following quadratic equations, leaving your answers in exact form.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{\left( 4x-5 \right)}^{2}}={{\left( 3-5{{x}^{2}} \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( {{y}^{2}}-y \right)}^{2}}-{{y}^{2}}+y=0$</div></li></ol>

Solve each of the following quadratic equations, leaving your answers in exact form.

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