Question

Resolva as equações a) x² + 9 x + 8 = 0 b) 9 x² - 24 x + 16 = 0 c) 3 x² - 15 x + 12 = 0 d) 10 x² + 72 x - 64 = 0​

Answer 1

$\boldsymbol{\red{\otimes}}$ Respostas:

a) S = { ( -8 , -1 ) }

b) S = { ( 4/3 ) }

c) S = { ( 1 , 4 ) }

d) S = { ( -8 , 4/5 ) }

$\boldsymbol{\red{\otimes}}$ Equações do 2 grau:

Para que possamos calcular uma equação do segundo grau devemos primeiramente identificar os termos ( a, b e c ) da sua equação, após isso devemos aplicar a fórmula de bhaskara.  Irei primeiramente identificar os termos ( a, b e c ) de cada uma das equações dadas. Veja:

Item a)

$\large\begin{array}{lr} \sf x^{2} + 9 x + 8 = 0\left\{\begin{array}{ll}\sf \boxed{\sf a=1}\\\sf \boxed{\sf b=9}\\\sf \boxed{\sf c = 8}\end{array}\right.\end{array}$

Item b)

$\large\begin{array}{lr} \sf 9 x^{2} - 24 x + 16 = 0\left\{\begin{array}{ll}\sf \boxed{\sf a=9}\\\sf \boxed{\sf b=-24}\\\sf \boxed{\sf c = 16}\end{array}\right.\end{array}$

Item c)

$\large\begin{array}{lr} \sf 3 x^{2} - 15 x + 12 = 0\left\{\begin{array}{ll}\sf \boxed{\sf a=3}\\\sf \boxed{\sf b=-15}\\\sf \boxed{\sf c = 12}\end{array}\right.\end{array}$

Item d)

$\large\begin{array}{lr} \sf 10 x^{2} + 72 x - 64 = 0 \left\{\begin{array}{ll}\sf \boxed{\sf a=10}\\\sf \boxed{\sf b=72}\\\sf \boxed{\sf c = -64}\end{array}\right.\end{array}$

Agora devemos aplicar a fórmula de bhaskara, veja abaixo à formula + a aplicação da mesma.

Item a)

$\large\begin{array}{lr} \sf x^{2} + 9 x + 8 = 0\\\\\sf \boxed{\sf a=1}\ \checkmark\\\sf \boxed{\sf b=9}\ \checkmark\\\sf \boxed{\sf c = 8}\ \checkmark\right.\end{array}$

$\large\begin{array}{lr}\sf x = \dfrac{-b\pm \sqrt{b^2 -4*a*c} }{2*a} \\\\\sf x = \dfrac{-9\pm \sqrt{9^2 -4*1*8} }{2*1} \\\\\sf x = \dfrac{-9\pm \sqrt{49} }{2}\\\\\sf x = \dfrac{-9\pm 7 }{2} \left\{\begin{array}{ll}\sf x' = \dfrac{-16}{2} \rightarrow \underline{\boxed{\sf -8}}\\\\\sf x'' = \dfrac{-2}{2} \rightarrow \underline{\boxed{\sf -1}} \end{array}\right.\end{array}$

Item b)

$\large\begin{array}{lr} \sf 9 x^{2} - 24 x + 16 = 0\\\\\sf \boxed{\sf a=9}\ \checkmark\\\sf \boxed{\sf b=-24}\ \checkmark\\\sf \boxed{\sf c = 16}\ \checkmark\right.\end{array}$

$\large\begin{array}{lr}\sf x = \dfrac{-b\pm \sqrt{b^2 -4*a*c} }{2*a} \\\\\sf x = \dfrac{-(-24)\pm \sqrt{(-24)^2 -4*9*16} }{2*9} \\\\\sf x = \dfrac{24\pm \sqrt{0} }{18}\\\\\sf x = \dfrac{24\pm 0 }{18} \left\{\begin{array}{ll}\sf x' = \dfrac{24}{18} \rightarrow \underline{\boxed{\sf \frac{4}{3} }}\\\\\sf x'' = \dfrac{24}{18} \rightarrow \underline{\boxed{\sf \frac{4}{3} }} \end{array}\right.\end{array}$

Item c)

$\large\begin{array}{lr} \sf 3 x^{2} - 15 x + 12 = 0\\\\\sf \boxed{\sf a=3}\ \checkmark\\\sf \boxed{\sf b=-15}\ \checkmark\\\sf \boxed{\sf c = 12}\ \checkmark\right.\end{array}$

$\large\begin{array}{lr}\sf x = \dfrac{-b\pm \sqrt{b^2 -4*a*c} }{2*a} \\\\\sf x = \dfrac{-(-15)\pm \sqrt{(-15)^2 -4*3*12} }{2*3} \\\\\sf x = \dfrac{15\pm \sqrt{81} }{6}\\\\\sf x = \dfrac{15\pm 9 }{6} \left\{\begin{array}{ll}\sf x' = \dfrac{6}{6} \rightarrow \underline{\boxed{\sf 1 }}\\\\\sf x'' = \dfrac{24}{6} \rightarrow \underline{\boxed{\sf 4 }} \end{array}\right.\end{array}$

Item d)

$\large\begin{array}{lr} \sf 10 x^{2} + 72 x - 64=0\\\\\sf \boxed{\sf a=10}\ \checkmark\\\sf \boxed{\sf b=72}\ \checkmark\\\sf \boxed{\sf c = -64}\ \checkmark\right.\end{array}$

$\large\begin{array}{lr}\sf x = \dfrac{-b\pm \sqrt{b^2 -4*a*c} }{2*a} \\\\\sf x = \dfrac{-72\pm \sqrt{(72)^{2} - 4*10*(-64)} }{2*10} \\\\\sf x = \dfrac{-72\pm \sqrt{7744} }{20}\\\\\sf x = \dfrac{-72\pm 88 }{20} \left\{\begin{array}{ll}\sf x' = \dfrac{-160}{20} \rightarrow \underline{\boxed{\sf -8 }}\\\\\sf x'' = \dfrac{16}{20} \rightarrow \underline{\boxed{\sf \frac{4}{5} }} \end{array}\right.\end{array}$

Espero ter ajudado.

Bons estudos.

• Att. FireClassis.

Answer 2

Resposta:

Resultado das equações

A) S = {-8,-1}

B) S = {4/3}

C) S = {1,4}

D) S = {-8,4/5}

Para resolvermos uma equação de segundo grau temos que passar por 3 etapas para achar as raízes.

• 1- etapa achar os coeficientes (a b e c). E muito importante para achar os coeficientes para podermos calcular o delta e o bhaskara.

• 2- etapa calcular o delta pela formula (-b² - 4ac). Precisamos achar o delta para a sua raiz ser usada na formula de bhaskara, e quando a sua raiz e negativa não existe soluções no conjunto dos números reais, apenas no conjunto dos números complexos.

• 3- etapa calcular o bhaskara pela formula $\frac{x=-b\pm\sqrt{\Delta} }{2a}$, quando caucularmos o delta, acharemos as raízes da equação

Resolução letra a)

x² + 9x + 8 = 0        

os coeficientes são   $a=1\\b=9\\c=8$

Delta

$\Delta=-b^{2} -4ac$

$\Delta=(-9)^{2} -4.1.8$

$\Delta=81-4.1.8$

$\Delta=81-32$

$\Delta=49$

bhaskara

$\frac{x=-b\pm\sqrt{\Delta} }{2a}$

Substituindo os valores de ( a, b e c ) na fórmula da bhaskara:

$\frac{x=-(+9)\pm\sqrt{49} }{2}$

$\frac{x=-9\pm7}{2}$

$x1=\frac{x=-9+7}{2} =-2/2=-1$

$x2=\frac{x=-9-7}{2} =-16/2=-8$

S = {-8,-1}

--------------------------------------------------------------------------------------------------------------

Letra B) 9x² - 24 x + 16 = 0

os coeficientes são   $a=9\\b=-24\\c=16$

Delta =

$\Delta=(-24)^{2} -4.9.16$

$\Delta=576-4.1.16$

$\Delta=576-576$

$\Delta=0$

Bhaskara

$\frac{x=24\pm0}{18}$

$x1=\frac{x=24+0}{18} =\frac{24}{18} :2=\frac{12}{9}:3=\frac{4}{3}$

$x2=\frac{x=24-0}{18} =\frac{24}{18} :2=\frac{12}{9}:3=\frac{4}{3}$

S = {4/3}

-------------------------------------------------------------------------------------------------------------

C) 3x² - 15 x + 12

Os coeficientes são  $a=3\\b=-15\\c=12$

Delta =

$\Delta=15^{2} -4.3.12$

$\Delta=225-4.3.12$

$\Delta=225-144$

$\Delta=81$

bhaskara

$\frac{x=15\pm9}{6}$

$x1=\frac{x=15+9}{6} =24/6=4$

$x2=\frac{x=15-9}{6} =6/6=1$

S = {1,4}

--------------------------------------------------------------------------------------------------------------

D) 10x² + 72 x - 64 = 0​

Coeficientes   $a=10\\b=72\\c=-64$

Delta

$\Delta=(-72)^{2} -4.10.(-64)$

$\Delta=5.184-4.10.(-64)$

$\Delta=5.184+2560$

$\Delta=7744$

Bhaskara

$\frac{x=-72\pm88}{20}$

$x1=\frac{x=-72+88}{20}=\frac{16}{20} :2=\frac{8}{10} :2=\frac{4}{5}$

$x2=\frac{-72-88}{20} =-160/20=-8$

S = {-8,4/5}