1.2

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> <FONT SIZE="4" COLOR="#00FFCC" FACE="Bell MT" ><B>P r e c á l c u l o 21</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B> |</B></FONT><FONT SIZE="3" COLOR="#00FFCC" FACE="Bell MT" ><B> </B></FONT><A HREF="index.htm" TARGET="_top" TITLE="Precálculo21"><U><B>Página principal</B></U></A><B>  |  </B><A HREF="id418.htm" TARGET="_top" TITLE="_"><U><B>Contenido</B></U></A><B>  | </B><A HREF="id492.htm" TARGET="_top" TITLE="Álgebra y trigonometría de Sullivan"><U><B>Álgebra y trigonometría de Sullivan</B></U></A><B> </B></div> <DIV ALIGN="LEFT"></DIV> <div> </div> </td> </tr><tr> <td valign="top" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="5543838917"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> <bR> </td> <td valign="top" height=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios 1.2</B></div> <div> <B>Ecuaciones cuadráticas</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="9294650114"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>Procedimiento </B></div> <div> Se van a utilizar cuatro métodos distintos para resolver ecuaciones cuadráticas (o de segundo grado), por factorización, método de la raíz cuadrada, completando cuadrados y aplicando la fórmula cudrática general.</div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>A.</B></FONT><FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B> </B></FONT> <B>Por factorización</B> (cuando es posible encontrar factores racionales):</div> <div> <B>1.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se hacen las restricciones pertinentes, evitando la división por 0</FONT></div> <div> <B>2</B><B>.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se escribe la ecuación en la forma estandar:</FONT></div> <div>      <IMG SRC="137a6b140.gif" border=0 width="107" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">.</div> <div> <B>3.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se factoriza el miembro izquierdo de la ecuación.</FONT></div> <DIV ALIGN="LEFT"></DIV> <div>  (<A HREF="http://www.calculo21.org/index.htm" TARGET="_top" TITLE="http://www.calculo21.org/index.htm"><U>En el álgebra de Baldor se estudian los casos de factorización en los Ejercicios 89 a 110, busque estos Ejercicios Aquí</U></A>)</div> <div> <B>4.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se igualan a cero cada uno de los dos factores obtenidos y se despeja la incógnita para cada factor.</FONT></div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>B.</B></FONT><FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B> </B></FONT> <B>Método de la raíz cuadrada</B> (cuando es posible encontrar factores racionales):</div> <div> <B>1.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se escribe la ecuación en la forma :</FONT></div> <div>      <IMG SRC="1394871d0.gif" border=0 width="135" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B>2.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se extrae raíz cuadrada en ambos miembros para obtener una ecuación equivalente de la forma:</FONT></div> <div> <IMG SRC="1395881b0.gif" border=0 width="136" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>3.  </B></FONT>Se despeja la <I>x.</I></div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>C.</B></FONT><FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B> </B></FONT> <B>Completando cuadrados</B>:</div> <div> <B>1.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se escribe la ecuación en la forma:</FONT></div> <div> <IMG SRC="13a354140.gif" border=0 width="84" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>2.  </B></FONT>Se divide cada término de la ecuación entre el coeficiente de x^2, esto es por <I>a</I>:</div> <div> <IMG SRC="13a454290.gif" border=0 width="84" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B>3.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se adiciona en ambos miembros de la ecuación el término que completa el trinomio cuadrado perfecto en el miembro izquierdo, que es de la forma:</FONT></div> <div> <IMG SRC="13a529310.gif" border=0 width="297" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B>4.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se factoriza el trinomio cuadrado perfecto en el miembro izquierdo:</FONT></div> <div> <IMG SRC="13a68c310.gif" border=0 width="140" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B>5.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se extrae raíz cuadrada en ambos miembros:</FONT></div> <div> <IMG SRC="13a78c310.gif" border=0 width="140" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>6.  </B></FONT>Se despeja la incógnita <I>x</I>:</div> <div> <IMG SRC="13a88d2f0.gif" border=0 width="141" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>7.  </B></FONT>Se calculan las dos raíces tomando una vez la fórmula con + y otra con <FONT SIZE="3" COLOR="#000000" FACE="Symbol" >-</FONT>.</div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>D.</B></FONT><FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B> </B></FONT> <B>Usando la fórmula cuadrática</B>:</div> <div> <B>1.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se escribe la ecuación en la forma estandar:</FONT></div> <div> <IMG SRC="13b554140.gif" border=0 width="84" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>2.  </B></FONT>Se deducem los valores numéricos de los coeficientes <I>a</I>, <I>b</I> y <I>c</I></div> <div> <B>3.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se calcula el valor del discriminante:</FONT></div> <div> <I><IMG SRC="13b83c140.gif" border=0 width="60" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> <div> <B>4.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Si el valor numérico del discriminante es mayor o igual a 0, la ecuación tiene soluciones reales y se procede con los siguientes pasos</FONT></div> <div> <B>5.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se sustituyen los valores numéricos de los coeficientes en la fórmula cuadrática:</FONT></div> <div> <I><IMG SRC="13ba832d0.gif" border=0 width="131" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> <div> <B>6.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se efectúan las operaciones indicadas y se simplifica</FONT></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>7.  </B></FONT>Se calculan las dos raíces tomando una vez la fórmula con + y otra con <FONT SIZE="3" COLOR="#000000" FACE="Symbol" >-</FONT>.</div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >9</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >28</FONT>, resuelva la ecuación por factorización:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="137b61140.gif" border=0 width="97" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#9" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13806b140.gif" border=0 width="107" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#11" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="138478140.gif" border=0 width="120" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#13" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="138688140.gif" border=0 width="136" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#16" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="138974170.gif" border=0 width="116" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#18" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="138c791b0.gif" border=0 width="121" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#20" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="138f67290.gif" border=0 width="103" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#25" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1392bf310.gif" border=0 width="191" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#27" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >29</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >34</FONT>, resuelva cada ecuación por el método de la raíz cuadrada:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="139855140.gif" border=0 width="85" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#29" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="139b6f1d0.gif" border=0 width="111" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#31" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="139f781d0.gif" border=0 width="120" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#34" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >35</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >40</FONT>, ¿qué número debe sumarse para completar el cuadrado de cada expresión:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="13a253140.gif" border=0 width="83" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#35" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13ab57290.gif" border=0 width="87" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#40" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >41</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >46</FONT>, resuelva cada ecuación completando cuadrados:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="13ae74140.gif" border=0 width="116" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#41" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13b190290.gif" border=0 width="144" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#43" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13b485290.gif" border=0 width="133" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#45" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >47</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >66</FONT>, encuentre las soluciones reales, si las hay, de cada ecuación. Utilice la fórmula cuadrática:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="13bd83140.gif" border=0 width="131" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#47" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13c089140.gif" border=0 width="137" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#51" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13c385170.gif" border=0 width="133" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#53" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13c671140.gif" border=0 width="113" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#55" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13c95b140.gif" border=0 width="91" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#58" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13cc81140.gif" border=0 width="129" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#59" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13cf87290.gif" border=0 width="135" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#62" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13d280290.gif" border=0 width="128" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#64" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13d5342b0.gif" border=0 width="820" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B><A HREF="#93" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13d83c170.gif" border=0 width="828" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#95" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13da633d0.gif" border=0 width="867" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B><A HREF="#96" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> <B><IMG SRC="13db7d700.gif" border=0 width="125" height="112" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="13e3593d0.gif" border=0 width="857" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#97" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <bR> <div> <B><IMG SRC="13ef48750.gif" border=0 width="840" height="117" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#99" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13e83c290.gif" border=0 width="572" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#109" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13eb4b290.gif" border=0 width="587" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#110" TITLE="1.2"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> <B> Soluciones</B></div> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> <B><IMG SRC="137dc9800.gif" border=0 width="713" height="128" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <B><IMG SRC="137fe4800.gif" border=0 width="740" height="128" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>13</div> <div> <B><IMG SRC="1382e3800.gif" border=0 width="739" height="128" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>16</div> <div> <B><IMG SRC="13856ce80.gif" border=0 width="876" height="232" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="18"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>18</div> <div> <B><IMG SRC="1388e4800.gif" border=0 width="740" height="128" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="20"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>20</div> <div> <B><IMG SRC="138beb9d0.gif" border=0 width="747" height="157" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>25</div> <div> <B><IMG SRC="138e45030.gif" border=0 width="837" height="259" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div> <B><IMG SRC="13913f2c0.gif" border=0 width="831" height="300" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>29</div> <div> <B><IMG SRC="1397fc680.gif" border=0 width="764" height="104" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="31"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>31</div> <div> <B><IMG SRC="139a41a50.gif" border=0 width="833" height="165" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="34"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>34</div> <div> <B><IMG SRC="139e61400.gif" border=0 width="865" height="320" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="35"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>35</div> <div> <B><IMG SRC="13a1392b0.gif" border=0 width="825" height="299" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="40"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>40</div> <div> <B><IMG SRC="13aa39650.gif" border=0 width="825" height="357" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="41"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>41</div> <div> <IMG SRC="13ad5b030.gif" border=0 width="859" height="259" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="43"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>43</div> <div> <IMG SRC="13b050b50.gif" border=0 width="848" height="437" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="45"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>45</div> <div> <IMG SRC="13b3bbf00.gif" border=0 width="699" height="496" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="47"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>47</div> <div> <IMG SRC="13bc13830.gif" border=0 width="787" height="387" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="51"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>51</div> <div> <IMG SRC="13bf3b900.gif" border=0 width="827" height="400" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="53"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>53</div> <div> <IMG SRC="13c27fe00.gif" border=0 width="639" height="224" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="55"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>55</div> <div> <IMG SRC="13c513950.gif" border=0 width="787" height="405" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="58"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>58</div> <div> <IMG SRC="13c813930.gif" border=0 width="787" height="403" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="59"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>59</div> <div> <IMG SRC="13cb13900.gif" border=0 width="787" height="400" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="62"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>62</div> <div> <IMG SRC="13ce64350.gif" border=0 width="868" height="565" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="64"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>64</div> <div> <IMG SRC="13d16cbb0.gif" border=0 width="876" height="443" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="93"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>93</div> <div> <IMG SRC="13d4344d0.gif" border=0 width="820" height="333" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="95"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>95</div> <div> <IMG SRC="13d73c700.gif" border=0 width="828" height="368" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="96"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>96</div> <div> <IMG SRC="13df661f0.gif" border=0 width="870" height="287" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="97"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> <IMG SRC="13e555570.gif" border=0 width="853" height="343" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">7</div> <bR> <div>  <A NAME="99"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" >99</FONT></div> <div> <IMG SRC="13ee60a50.gif" border=0 width="864" height="421" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="109"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>109</div> <div> <IMG SRC="13e7f75d0.gif" border=0 width="759" height="349" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="110"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>110</div> <div> <IMG SRC="13ead79c0.gif" border=0 width="727" height="412" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> </td> </tr></table></body>