Question

Como resolver a equação 2x²-6x+4=0 com o quadrado perfeito?

Answer 1

$2x^2-6x+4~=~0\\\\\\Simplificando~a~equacao~por~2\\\\\\x^2-3x+2~=~0\\\\\\Vamos~agora~completar~o~quadrado\\\\\\(x\,+\alpha)^2+\beta~=~x^2-3x+2\\\\\\x^2+2x\alpha+\alpha^2+\beta~=~x^2-3x+2\\\\\\\left\{\begin{array}{ccc}2\alpha&=&-3\\\alpha^2+\beta&=&2\end{array}\right\\\\\\\alpha~=~\frac{-3}{2}~=~\boxed{-\frac{3}{2}~~ou~~-1,5}\\\\\\\alpha^2+\beta~=~2\\\\\\\left(-\frac{3}{2}\right)^2+\beta~=~2\\\\\\\beta~=~2-\frac{9}{4}\\\\\\\boxed{\beta~=~-\frac{1}{4}~~ou~~-0,25}$

$A~equacao~fica:\\\\\\\left(x-\frac{3}{2}\right)^2-\frac{1}{4}~=~0\\\\\\\left(x-\frac{3}{2}\right)^2~=~\frac{1}{4}\\\\\\x-\frac{3}{2}~=~\pm\sqrt{\frac{1}{4}}\\\\\\x~=~\pm\frac{1}{2}+\frac{3}{2}\\\\\\x'~=~\frac{1}{2}+\frac{3}{2}~=~\boxed{2}\\\\\\x''~=~-\frac{1}{2}+\frac{3}{2}~=~\boxed{1}$