Question

2x² + 2x + 5 = (4x - 1)√(x² + 3 )

Answer 1

Pt đã cho \(\Leftrightarrow2\left(x^2+3\right)-\left(4x-1\right)\sqrt{x^2+3}+2x-1=0\)

Đặt \(t=\sqrt{x^2+3};\left(t>0\right)\) pt trở thành:

\(2t^2-\left(4x-1\right)t+2x-1=0\)

\(\Delta_t=\left(4x-1\right)^2-4\cdot2\cdot\left(2x-1\right)=16x^2-24x+9=\left(4x-3\right)^2\)

\(t_1=\frac{4x-1+4x-3}{4}=2x-1\) \(\Leftrightarrow\sqrt{x^2+3}=2x-1\Leftrightarrow x=\frac{2+\sqrt{10}}{3}\) (vì \(x>\frac{1}{2}\))

\(t_2=\frac{4x-1-4x+3}{4}=\frac{1}{2}\) (loại)

Vậy pt đã cho có nghiệm \(x=\frac{2+\sqrt{10}}{3}\)