Question

Resolva essa equação irracional.
$\sqrt[]{x} {}^{2} - x + 4 = 4$
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Answer 1

$\underbrace{Veja:}$

$\Rightarrow~~\sf \sqrt{x^2-x+4}=4$

> para tirar a raiz quadrada, eleve os dois membros ao quadrado

$\Rightarrow~~\sf (\sqrt{x^2-x+4})^2=(4)^2$

$\Rightarrow~~\sf x^2-x+4=16$

$\Rightarrow~~\sf x^2-x+4-16=0$

$\Rightarrow~~\sf x^2-x-12=0$

> surgiu uma equação do 2ºgrau, vamos resolver aplicando a fórmula de Bhaskara

> coeficientes:

• a = 1
• b = -1
• c = -12

$~~$

$\sf \Delta=b^2-4ac$

$\sf \Delta=(-1)^2-4.(1).(-12)$

$\sf \Delta=1+48$

$\sf \Delta=49$

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

$\sf x=\dfrac{-(-1)\pm\sqrt{49}}{2.(1)}~~\Rightarrow~~x=\dfrac{1\pm7}{2}$

$\sf x'=\dfrac{1+7}{2}~~\Rightarrow~~x'=\dfrac{8}{2}~~\Rightarrow~~\boxed{\sf x'=4}$

$\sf x''=\dfrac{1-7}{2}~~\Rightarrow~~x''=-\dfrac{6}{2}~~\Rightarrow~~\boxed{\sf x''=-3}$

$~~$

Resposta: o conjunto solução é:

          $\boxed{\sf S=\left\{-3~~;~~4\right\}}$