Question

Resolva a equação x²+10/3 x+1=0 utilizando o método de completar os quadrados

Answer 1

Olá!

$\displaystyle \\ \mathsf{x^2 + \frac{10x}{3} + 1 = 0} \\\\\\ \mathsf{x^2 + 2 \cdot \frac{5x}{3} + 1 = 0} \\\\\\ \mathsf{\left ( x^2 + 2 \cdot x \cdot \frac{5}{3} + \frac{25}{9} \right ) - \frac{25}{9} + 1 = 0} \\\\\\ \mathsf{\left ( x + \frac{5}{3} \right )^2 = \frac{25}{9} - 1} \\\\\\ \mathsf{\left ( x + \frac{5}{3} \right )^2 = \frac{16}{9}} \\\\\\ \mathsf{\left(x+\frac{5}{3}\right ) = \pm \frac{4}{3}} \\\\\\ \Rightarrow \mathsf{x_1 + \frac{5}{3} = \frac{4}{3} \Leftrightarrow \boxed{\mathsf{x_1 = - \frac{1}{3}}}} \\\\\\ \Rightarrow \mathsf{x_2 + \frac{5}{3} = - \frac{4}{3} \Leftrightarrow \mathsf{x_2 = - \frac{4}{3} - \frac{5}{3}}\Leftrightarrow \boxed{\mathsf{x_2 = - 3}}}$