Question

(PUC-SP) O conjunto verdade da equação raiz 4x 1=2x-1 é?

Answer 1

$\sqrt{4x+1}=2x-1\\ ( \sqrt{4x+1})^2=(2x-1)^2\\ 4x+1=(2x)^2-2\cdot(2x)\cdot1+(1)^2\\ 4x+1=4x^2-4x+1\\ 4x^2-4x-4x+1-1=0\\ 4x^2-8x=0~~(divide~por~4)\\ x^2-2x=0\\ x(x-2)=0\\\\ x_1=0\\ x-2=0\Rightarrow x_2=2$

Agora testaremos as raízes $\sqrt{4x+1}=2x-1$..

$\sqrt{4\cdot0+1}=2\cdot0-1\\ \sqrt{1}=-1~~(\text{falso})$

$\sqrt{4\cdot2+1}=2\cdot2-1\\ \sqrt{8+1}=4-1\\ \sqrt{9}=3~~(\text{verdadeiro})$

Portanto..


$\Large\boxed{\text{S}=\{\text{2\}}}$