tag:blogger.com,1999:blog-5580933574788045792024-02-20T11:45:27.143-08:00EQUAÇÕES INCOMPLETAS DO 2º GRAUUnknownnoreply@blogger.comBlogger1125tag:blogger.com,1999:blog-558093357478804579.post-13429928269987386002011-11-09T05:04:00.000-08:002011-11-09T05:04:52.244-08:00EQUAÇÕES INCOMPLETAS DO 2º GRAU<div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="color: red; font-size: large;">EQUAÇÕES INCOMPLETAS DO 2º GRAU</span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;">Resolver uma equação é determinar todas as suas soluções. Vejamos, através de exemplos, como se resolvem as equações incompletas do 2º grau</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="color: blue; font-size: large;">1º CASO</span>:<span style="mso-spacerun: yes;"> </span>Equação da forma <u><em><span style="color: purple;">ax² + c = 0</span></em></u></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;">Exemplos:</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;">Resolver as seguintes equações, sendo U = R</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l3 level1 lfo1; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">1)<span style="font-family: "Times New Roman";"> </span></span>x² - 25 = 0</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x² = 25</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x = + ou - √ 25</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x = + ou – 5</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">Logo : V = { +5, -5}</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l3 level1 lfo1; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">2)<span style="font-family: "Times New Roman";"> </span></span>2x² - 18 = 0</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">2x²= 18</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x² = 18 / 2</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x² = 9</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x = + ou - √9</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x = + ou – 3</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">Logo V = { +3, -3}</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l3 level1 lfo1; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">3)<span style="font-family: "Times New Roman";"> </span></span>7x²- 14 = 0</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">7x²= 14</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x²= 14/ 7</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x² = 2</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x = + ou - √2 </div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">Logo V = { +√2, -√2}</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l3 level1 lfo1; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">4)<span style="font-family: "Times New Roman";"> </span></span>x ²+ 25 = 0</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x²= -25</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">x =<span style="mso-spacerun: yes;"> </span>+ ou - √-25 = nenhum real, pois (nenhum real)² = -25</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">Logo V = vazio</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;">EXERCÍCIOS </div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">1)<span style="font-family: "Times New Roman";"> </span></span>Resolva as seguintes equações do 2º grau , sendo U = R</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">a)<span style="font-family: "Times New Roman";"> </span></span>x²- 49 = 0<span style="mso-spacerun: yes;"> </span>(R: 7, -7)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">b)<span style="font-family: "Times New Roman";"> </span></span>x² = 1<span style="mso-spacerun: yes;"> </span>(R: 1, -1)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">c)<span style="font-family: "Times New Roman";"> </span></span>2x² - 50 = 0 (R: 5, -5)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">d)<span style="font-family: "Times New Roman";"> </span></span>7x² - 7 = 0 (R: 1, -1)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">e)<span style="font-family: "Times New Roman";"> </span></span>4x²= 36 (R: 3, -3)<span style="mso-spacerun: yes;"> </span>( marcar)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">f)<span style="font-family: "Times New Roman";"> </span></span>5x² - 15 = 0 (R: √3, -√3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">g)<span style="font-family: "Times New Roman";"> </span></span>21 =<span style="mso-spacerun: yes;"> </span>7x²<span style="mso-spacerun: yes;"> </span>(R: √3, -√3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">h)<span style="font-family: "Times New Roman";"> </span></span>5x² + 20 = 0 (R: vazio)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">i)<span style="font-family: "Times New Roman";"> </span></span>4x² - 49 = 0 ( R: 7/2, -7/2) </div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">j)<span style="font-family: "Times New Roman";"> </span></span>16 = 9x² (R: 4/3 , -4/3) </div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">k)<span style="font-family: "Times New Roman";"> </span></span>3x² + 30 = 0 (R: vazio)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">l)<span style="font-family: "Times New Roman";"> </span></span>9x² - 5 = 0 (R: √5/3 , -√5/3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">2)<span style="font-family: "Times New Roman";"> </span></span>Resolva as equações do 2º grau, sendo U = R</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l2 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">a)<span style="font-family: "Times New Roman";"> </span></span>7x² + 2 = 30 (R: 2, -2)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l2 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">b)<span style="font-family: "Times New Roman";"> </span></span>2x² - 90 = 8 (R: 7, -7)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l2 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">c)<span style="font-family: "Times New Roman";"> </span></span>4x²- 27 = x² (R: 3, -3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l2 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">d)<span style="font-family: "Times New Roman";"> </span></span>8x² = 60 – 7x² (R: 2, -2)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">3)<span style="font-family: "Times New Roman";"> </span></span>Resolva as equações do 2º grau, sendo U = R</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l4 level1 lfo5; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">a)<span style="font-family: "Times New Roman";"> </span></span>3 (x² - 1) = 24 (R: 3, -3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l4 level1 lfo5; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">b)<span style="font-family: "Times New Roman";"> </span></span>2( x² - 1) = x² + 7 (R: 3, -3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l4 level1 lfo5; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">c)<span style="font-family: "Times New Roman";"> </span></span>5(x² - 1) = 4(x² + 1) (R: 3, -3)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 36pt; mso-list: l4 level1 lfo5; tab-stops: list 36.0pt; text-indent: -18pt;"><span style="mso-list: Ignore;">d)<span style="font-family: "Times New Roman";"> </span></span>(x -3) (x – 4) + 8 = x<span style="mso-spacerun: yes;"> </span>(R: 2, -2)</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div><div class="MsoNormal" style="margin: 0cm 0cm 0pt 18pt;"><span style="font-family: "Trebuchet MS"; font-size: 18pt;"><span style="color: blue;">2° CASO</span></span><span style="font-family: "Trebuchet MS";">: Equações da forma <span style="color: purple;"><em><u><span style="background: white; font-family: "Trebuchet MS";">ax² + bx = 0 ( c = 0)</span></u></em></span><br />
Propriedade: Para que um produto seja nulo é preciso que um dos fatores seja zero .<br />
<br />
Exemplos<br />
<br />
1) resolver x² - 5x = 0<br />
fatorando x ( x – 5) = 0<br />
<br />
deixando um dos fatores nulo temos x = 0<br />
<br />
e o outro x – 5 = 0 , passando o 5 para o outro lado do igual temos x = 5<br />
<br />
logo V= (0 e 5)<br />
<br />
2) resolver: 3x² - 10x = 0<br />
fatorando: x (3x – 10) = 0<br />
<br />
deixando um dos fatores nulo temos x = 0<br />
<br />
Tendo também 3x – 10 = 0<br />
3x = 10<br />
x = 10/3<br />
<br />
logo V= (0 e 10/3)<br />
<br />
Observe que, nesse caso, uma das raízes é sempre zero.<br />
<br />
<br />
EXERCÍCIOS<br />
<br />
1) Resolva as seguintes equações do 2° grau.<br />
<br />
a) x² - 7x = 0 (R: 0 e 7)<br />
b) x² + 5x = 0 (R: 0 e -5)<br />
c) 4x² - 9x = 0 (R: 0 e 9/4)<br />
d) 3x² + 5x =0 (R: 0 e -5/3)<br />
e) 4x² - 12x = 0 (R: 0 e 3)<br />
f) 5x² + x = 0 (R: 0 e -1/5)<br />
g) x² + x = 0 (R: 0 e -1)<br />
h) 7x² - x = 0 (R: 0 e 1/7)<br />
i) 2x² = 7x (R: 0 e 7/2)<br />
j) 2x² = 8x (R: 0 e 4)<br />
k) 7x² = -14x (R: 0 e -2)<br />
l) -2x² + 10x = 0 (R: 0 e 5)<br />
<br />
2) Resolva as seguintes equações do 2° grau<br />
<br />
a) x² + x ( x – 6 ) = 0 (R: 0 e 3)<br />
b) x(x + 3) = 5x (R: 0 e 2)<br />
c) x(x – 3) -2 ( x-3) = 6 (R: 0 e 5)<br />
d) ( x + 5)² = 25 (R: 0 e -10)<br />
e) (x – 2)² = 4 – 9x (R: 0 e -5)<br />
f) (x + 1) (x – 3) = -3 (R: 0 e 2)</span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br />
</div>Unknownnoreply@blogger.com30