Đặt \(P\left(x\right)=\frac{3(x-\sqrt{3})(x-\sqrt{5})}{(1-\sqrt{3})(1-\sqrt{5})}+\frac{4(x-1)(x-\sqrt{5})}{(\sqrt{3}-1)(\sqrt{3}-\sqrt{5})}+\frac{5(x-1)(x-\sqrt{3})}{(\sqrt{5}-1)(\sqrt{5}-\sqrt{3})}\)

Khi x=1 thì \(P\left(1\right)=\frac{3(1-\sqrt{3})(1-\sqrt{5})}{(1-\sqrt{3})(1-\sqrt{5})}+\frac{4(1-1)(1-\sqrt{5})}{(\sqrt{3}-1)(\sqrt{3}-\sqrt{5})}+\frac{5(1-1)(1-\sqrt{3})}{(\sqrt{5}-1)(\sqrt{5}-\sqrt{3})}\) =3

Khi x=\(\sqrt3\) thì \(P\left(\sqrt3\right)=\frac{3(\sqrt3-\sqrt{3})(\sqrt3-\sqrt{5})}{(1-\sqrt{3})(1-\sqrt{5})}+\frac{4(\sqrt3-1)(\sqrt3-\sqrt{5})}{(\sqrt{3}-1)(\sqrt{3}-\sqrt{5})}+\frac{5(\sqrt3-1)(\sqrt3-\sqrt{3})}{(\sqrt{5}-1)(\sqrt{5}-\sqrt{3})}=4\)

Khi x=\(\sqrt5\) thì \(P\left(\sqrt5\right)=\frac{3(\sqrt5-\sqrt{3})(\sqrt5-\sqrt{5})}{(1-\sqrt{3})(1-\sqrt{5})}+\frac{4(\sqrt5-1)(\sqrt5-\sqrt{5})}{(\sqrt{3}-1)(\sqrt{3}-\sqrt{5})}+\frac{5(\sqrt5-1)(\sqrt5-\sqrt{3})}{(\sqrt{5}-1)(\sqrt{5}-\sqrt{3})}\) =5

Đặt \(P\left(x\right)=a\cdot x^2+bx+c\)

Vì P(x) đi qua các điểm A(1;3); B\(\left(\sqrt3;4\right)\) ; \(C\left(\sqrt5;5\right)\) nên ta sẽ có:

\(\begin{cases}a\cdot1^2+b\cdot1+c=3\\ a(\sqrt{3})^2+b\sqrt{3}+c=4\\ a(\sqrt{5})^2+b\sqrt{5}+c=5\end{cases}\iff\begin{cases}a+b+c=3\quad(1)\\ 3a+\sqrt{3}b+c=4\quad(2)\\ 5a+\sqrt{5}b+c=5\quad(3)\end{cases}\)

=>a=1/2; b=0; c=5/2

=>\(P(x) = \frac{1}{2}x^2 + \frac{5}{2}\)

P(x)=3x

=>\(\frac12x^2+\frac52=3x\)

=>\(0,5x^2-3x+2,5=0\)

=>\(x^2-6x+5=0\)

=>(x-5)(x-1)=0

=>x=5 hoặc x=1

Tham khảo:

1) Giải phương trình : \(11\sqrt{5-x}+8\sqrt{2x-1}=24+3\sqrt{\left(5-x\right)\left(2x-1\right)}\) - Hoc24

\(\sqrt{x+3}+2\sqrt{x}=2+\sqrt{x\left(x+3\right)}\left(đk:x\ge0\right)\)

\(\Leftrightarrow x+3+4x+4\sqrt{x\left(x+3\right)}=4+x\left(x+3\right)+4\sqrt{x\left(x+3\right)}\)

\(\Leftrightarrow5x+3=4+x^2+3x\)

\(\Leftrightarrow x^2-2x+1=0\)

\(\Leftrightarrow\left(x-1\right)^2=0\)

\(\Leftrightarrow x-1=0\)

\(\Leftrightarrow x=1\left(tm\right)\)

Lấy phương trình trên trừ phương trình dưới thu được:

\(2\left(y-x\right)=-2\Rightarrow y=x-1\)

Thay vào phương trình dưới suy ra:

\(2\sqrt{2}x=4\sqrt{2}0\Rightarrow x=2\Rightarrow y=1\)