Đáp án + giải thích các bước giải:
1/ `\sqrt{5-3x}+\sqrt{x+1}=\sqrt{3x^2-4x+4} (-1<=x<=5/3)`
`->5-3x+x+1+2\sqrt{(5-3x)(x+1)}=3x^2-4x+4`
`->2\sqrt{-3x^2+2x+5}=3x^2-2x-2`
Đặt `a=\sqrt{-3x^2+2x+5} (a>=0)`, phương trình trở thành
`2a=-a^2+3`
`->a^2+2a-3=0`
`->a^2+2a+1-4=0`
`->(a+1)^2=4`
`->a+1=2` `(`do `a>=0->a+1>0)`
`->a=1`
`->\sqrt{-3x^2+2x+5}=1`
`->-3x^2+2x+5=1`
`->3x^2-2x-4=0`
`->9x^2-6x-12=0`
`->(3x)^2-2.3x+1-13=0`
`->(3x-1)^2=13`
`->`\(\left[ \begin{array}{l}3x-1=\sqrt{13}\\3x-1=-\sqrt{13}\end{array} \right.\)
`->`\(\left[ \begin{array}{l}x=\dfrac{\sqrt{13}+1}{3}\\x=\dfrac{-\sqrt{13}+1}{3}\end{array} (TM) \right.\)
2/ `x^3-x^2-12x\sqrt{x-1}+20=0 (x>=1)`
`->x^2(x-1)-12x\sqrt{x-1}+36-16=0`
`->(x\sqrt{x-1}-6)^2=16`
`->`\(\left[ \begin{array}{l}x\sqrt{x-1}-6=4\\x\sqrt{x-1}-6=-4\end{array} \right.\)
`->`\(\left[ \begin{array}{l}x\sqrt{x-1}=10\\x\sqrt{x-1}=-2 \end{array} \right.\)
`->`\(\left[ \begin{array}{l}x^2(x-1)=100\\x^2(x-1)=4\end{array} \right.\)
`->`\(\left[ \begin{array}{l}x^3-5x^2+4x^2-20x+20x-100=0\\x^3-2x^2+x^2-2x+2x-4=0\end{array} \right.\)
`->`\(\left[ \begin{array}{l}(x-5)(x^2+4x+20)=0\\(x-2)(x^2+x+2)=0\end{array} \right.\)
`->`\(\left[ \begin{array}{l}x=5\\x=2\end{array}(TM) \right.\)
