a)\(1+\sqrt{3x+1}=3x\)\(\Leftrightarrow\sqrt{3x+1}=3x-1\Leftrightarrow3x+1=\left(3x-1\right)^2\)
\(\Leftrightarrow3x-1=9x^2-6x+1\Leftrightarrow9x^2-6x+1-3x+1=0\)
\(\Leftrightarrow9x^2-9x+2=0\Leftrightarrow9x^2-6x-3x+2=0\)
\(\Leftrightarrow3x\cdot\left(3x-2\right)-\left(3x-2\right)=0\Leftrightarrow\left(3x-1\right)\left(3x-2\right)=0\)
\(\Leftrightarrow\left[\begin{array}{nghiempt}3x-1=0\\3x-2=0\end{cases}\Leftrightarrow\left[\begin{array}{nghiempt}x=\frac{1}{3}\\x=\frac{2}{3\left[\right]}\end{array}\right.}\)
b. \(\frac{\sqrt{5x+7}}{x+3}=4\)
ĐKXĐ: \(x\ge-\frac{7}{5}\)
\(\Leftrightarrow\sqrt{5x+7}=4\left(x+3\right)\\ \Leftrightarrow\left(\sqrt{5x+7}\right)^2=\left[4\left(x+3\right)\right]^2\\ \Leftrightarrow5x+7=16\left(x^2+6x+9\right)\\ \Leftrightarrow5x+7=16x^2+96x+144\\ \Leftrightarrow16x^2+96x-5x+144-7=0\\ \Leftrightarrow16x^2+91x+137=0\\ \Leftrightarrow\left(4x\right)^2+2.4x.\frac{91}{8}+\frac{8281}{64}+\frac{487}{64}=0\\ \Leftrightarrow\left(4x+\frac{91}{8}\right)^2+\frac{487}{64}=0\left(1\right)\)
Mà \(\left(4x+\frac{91}{8}\right)^2\ge0\forall x\Rightarrow\left(4x+\frac{91}{8}\right)^2+\frac{487}{64}\ge\frac{487}{64}>0\forall x\)
\(\Rightarrow\) phương trình (1) không xảy ra.
Vậy không cógiá trị nào của x thỏa mãn phương trình.
