Question

Resolva as raízes irracionais : U = R ​

Answer 1

Ola!!

Dada as equações irracionais:

${\color{blue}{x-3=2\sqrt{x}}}$

$2\sqrt{x}=x-3$

$\sqrt{x}=\frac{x-3}{2}$

${\color{blue}{x=(\frac{x-3}{2})^2}}$

$x=\frac{(x-3)^2}{2^2}$

$x=\frac{x^2-2*x*3+3^2}{4}$

${\color{blue}{x=\frac{x^2-6x+9}{4}}}$

$x*4=x^2-6x+9$

${\color{blue}{4x=x^2-6x+9}}$

$x^2-6x-4x+9=0$

${\color{blue}{x^2-10x+9=0}}$

$Coeficientes:\left\{\begin{array}{cc}a=1\\b=-10\\c=9\\\end{array}\right$

Lembrando que:

$\large\boxed{\boxed{{x_{1,2}=\frac{-b\pm\sqrt{b^2-4*a*c}}{2*a}}}}}}$

$x_{1,2}=\frac{10\pm\sqrt{(-10)^2-4*1*9}}{2*1}$

$x_{1,2}=\frac{10\pm\sqrt{100-36}}{2}=\frac{10\pm\sqrt{64}}{2}$

${\color{blue}{x_{1}=\frac{10+8}{2}=\frac{18}{2}=9}}$

${\color{blue}{x_{2}=\frac{10-8}{2}=\frac{2}{2}=1}}$

${\color{blue}{Sol:\{\:1;\:9}}}$

B)

${\color{blue}{→2x=\sqrt{9x-2}}}$

$9x-2=(2x)^2$

$9x-2=4x^2$

${\color{blue}{-4x^2+9x-2=0}}$

$Coeficientes:\left\{\begin{array}{cc}a=-4\\b=9\\c=-2\\\end{array}\right$

Achando primeiro seu Discriminante pela expressão:

$\Delta=b^2-4*a*c$

$\Delta=9^2-4*(-4)*(-2)$

$\Delta=81-32$

${\color{blue}{\Delta=49}}$

Achando as raízes pelo Bhaskara:

$\large\boxed{\boxed{{x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2*a}}}}}}$

$x_{1}=\frac{-9+\sqrt{49}}{2*(-4)}=\frac{-9+7}{-8}$

${\color{blue}{x_{1}=\frac{-2}{-8}=\frac{1}{4}}}$

$x_{2}=\frac{-9-7}{2*(-4)}=\frac{-16}{-8}$

${\color{blue}{x_{2}=2}}$

$Sol:\{\:\frac{1}{4};2\:\}$

Espero ter ajudado bastante;)