Question

Giải phương trình

x²-11x-9+3căn(x³+1)=0

\(x^2-11x-9+3\sqrt{x^3+1}=0\)

Answer 1

DK \(x^3+1\ge0\Leftrightarrow\left(x+1\right)\left(x^2-x+1\right)\ge0\Leftrightarrow x\ge-1\)

ta thay x=-1 ko phai la nghiem => x>-1

pt <=> \(\left(x^2-5x-3\right)+3\left(\sqrt{x^3+1}-2\left(x+1\right)\right)=0\)

<=> \(\left(x^2-5x-3\right)+3\left(\frac{x^3+1-4x^2-8x-4}{\sqrt{x^3+1}+2\left(x+1\right)}\right)=0\)

<=> \(x^2-5x-3+3\left[\frac{\left(x+1\right)\left(x^2-5x+3\right)}{\sqrt{x^3+1}+2\left(x+1\right)}\right]=0\)

<=> \(\left(x^2-5x-3\right)\left(1+\frac{3\left(x+1\right)}{\sqrt{x^3+1}+2\left(x+1\right)}\right)=0\)

<=> x^2 -5x-3=0 ( do cai trong ngoac thu 2 vo nghiem vi X>-1)

<=> \(x=\frac{5\pm\sqrt{37}}{2}\) tmdk

Vay \(S=\left\{\frac{5-\sqrt{37}}{2};\frac{5+\sqrt{37}}{2}\right\}\)