Question

Resolva a equação
$x + \sqrt{3x + 7} = 1$
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Answer 1

$\text x + \sqrt{3\text x+7} = 1$

isola a raiz quadrada :

$\sqrt{3\text x+7} = 1 - \text x$

elevando ao quadrado dos dois lados :

$3\text x+7 = (1 - \text x)^2$

$|3\text x+7| = 1-2\text x + \text x^2$

$\text x^2 - 5\text x -6 = 0$

$\displaystyle \text x = \frac{-(-5)\pm\sqrt{(-5)^2-4.1.(-6)}}{2}$

$\displaystyle \text x = \frac{5\pm\sqrt{25+24}}{2} \to \displaystyle \text x = \frac{5\pm\sqrt{49}}{2} \to \displaystyle \text x = \frac{5\pm7}{2}$

$\displaystyle \text x = \frac{5+7}{2} \to \text x = 6}$

$\displaystyle \text x = \frac{5-7}{2} \to \text x = - 1}$

Verificando as soluções :

x = 6

$\text x+\sqrt{3\text x+7} = 1 \to 6+\sqrt{3.6+7} = 1 \to 6+5 = 1$ (FALSO )

x = -1

$\text x+\sqrt{3\text x+7} = 1 \to -1+\sqrt{3.(-1)+7} = 1 \to -1+2 = 1$ ( VERDADEIRO)

Solução :

$\huge\boxed{\text x = -1}$