Question

Determine as raízes das equações do 2º grau a seguir: a) x²-12x-13=0. b) 2 x²+2x-24=0. c) x²+5x-24=0

Answer 1

• Resposta:
  
  a)
  
  $x^2-12x-13=0\\1*x^2+-12*x=13\\1*x^2+-12*x+(-6)^2=-6^2+13\\1*x^2+-12*x+(-6)^2=36+13\\1*x^2+-12*x+(-6)^2=36+13\\1*(1*x+(-6))^2=49\\1*x+(-6)=+-*49^0.5\\1*x_1+(-6)=49^0.5\\1*x_1+-6=49^0.5\\1*x_1+-6=7\\1*x_1=13\\1*x_2+(-6)=-1*49^0.5\\1*x_2+-6=-1*49^0.5\\1*x_2+-6=-1*7\\1*x_2=-1\\Conjunto solucao: {13;-1}$  
  
  b)
  
  $2*x^+2x-24=0\\2*x^2+2*x=24\\1*x^2+1*x=12\\1*x^2+1*x+(1/2)^2=1/2^2+12\\1*x^2+1*x+(1/2)^2=1/4+12\\1*x^2+1*x+(1/2)^2=1*1/4+12\\1*x^2+1*x+(1/2)^2=1*1/4+1*48/4\\1*(1*x+(1/2))^2=1*49/4\\1*x+(1/2)=+-\sqrt{49/4\\1*x_1+(1/2)=-\sqrt{49/4\\1*x_1+0.5=-\sqrt{49/4\\1*x_1+0.5=-\sqrt{12.25\\1*x_1+0.5=3.5\\1*x_1=3\\1*x_2+(1/2)=-\sqrt{49/4\\1*x_2+0.5=-\sqrt{49/4\\1*x_2+0.5=-\sqrt{12.25\\1*x_2+0.5=-1*3.5\\1*x_2=-4\\Conjunto solucao: {3;-4}$
  
  c)
  
  $1*x^2+5x-24=0\\1*x^2+5*x=24\\1*x^2+5*x+(5/2)^2=5/2^2+24\\1*x^2+5*x+(5/2)^2=25/4+24\\1*x^2+5*x+(5/2)^2=1*25/4+24\\1*x^2+5*x+(5/2)^2=1*25/4+1*96/4\\1*(1*x+(5/2))^2=1*121/4\\1*x+(5/2)=+-\sqrt{121/4\\1*x_1+(5/2)=-\sqrt{121/4\\1*x_1+2.5=-\sqrt{121/4\\1*x_1+2.5=-\sqrt{30.25\\1*x_1+2.5=5.5\\1*x_1=3\\1*x_2+(5/2)=-\sqrt{121/4\\1*x_2+2.5=-\sqrt{121/4\\1*x_2+2.5=-\sqrt{30.25\\1*x_2+2.5=-1*5.5\\1*x_2=-8\\Conjunto solucao: {3;-8}$
  

Explicação passo a passo:

• 
  a)
  
  $x^2-12x-13=0\\1*x^2+-12*x=13\\1*x^2+-12*x+(-6)^2=-6^2+13\\1*x^2+-12*x+(-6)^2=36+13\\1*x^2+-12*x+(-6)^2=36+13\\1*(1*x+(-6))^2=49\\1*x+(-6)=+-*49^0.5\\1*x_1+(-6)=49^0.5\\1*x_1+-6=49^0.5\\1*x_1+-6=7\\1*x_1=13\\1*x_2+(-6)=-1*49^0.5\\1*x_2+-6=-1*49^0.5\\1*x_2+-6=-1*7\\1*x_2=-1\\Conjunto solucao: {13;-1}$  
  
  b)
  
  $2*x^+2x-24=0\\2*x^2+2*x=24\\1*x^2+1*x=12\\1*x^2+1*x+(1/2)^2=1/2^2+12\\1*x^2+1*x+(1/2)^2=1/4+12\\1*x^2+1*x+(1/2)^2=1*1/4+12\\1*x^2+1*x+(1/2)^2=1*1/4+1*48/4\\1*(1*x+(1/2))^2=1*49/4\\1*x+(1/2)=+-\sqrt{49/4\\1*x_1+(1/2)=-\sqrt{49/4\\1*x_1+0.5=-\sqrt{49/4\\1*x_1+0.5=-\sqrt{12.25\\1*x_1+0.5=3.5\\1*x_1=3\\1*x_2+(1/2)=-\sqrt{49/4\\1*x_2+0.5=-\sqrt{49/4\\1*x_2+0.5=-\sqrt{12.25\\1*x_2+0.5=-1*3.5\\1*x_2=-4\\Conjunto solucao: {3;-4}$
  
  c)
  
  $1*x^2+5x-24=0\\1*x^2+5*x=24\\1*x^2+5*x+(5/2)^2=5/2^2+24\\1*x^2+5*x+(5/2)^2=25/4+24\\1*x^2+5*x+(5/2)^2=1*25/4+24\\1*x^2+5*x+(5/2)^2=1*25/4+1*96/4\\1*(1*x+(5/2))^2=1*121/4\\1*x+(5/2)=+-\sqrt{121/4\\1*x_1+(5/2)=-\sqrt{121/4\\1*x_1+2.5=-\sqrt{121/4\\1*x_1+2.5=-\sqrt{30.25\\1*x_1+2.5=5.5\\1*x_1=3\\1*x_2+(5/2)=-\sqrt{121/4\\1*x_2+2.5=-\sqrt{121/4\\1*x_2+2.5=-\sqrt{30.25\\1*x_2+2.5=-1*5.5\\1*x_2=-8\\Conjunto solucao: {3;-8}$