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\(\sqrt{5+3x}-\sqrt{5-3x}=a\left(x\le\dfrac{5}{3}\right)\)
\(\Rightarrow\left(\sqrt{5+3x}-\sqrt{5-3x}\right)^2=a^2\)
\(\Rightarrow5+3x+5-3x-2\sqrt{\left(5+3x\right)\left(5-3x\right)}=a^2\)
\(\Rightarrow10-2\sqrt{25-9x^2}=a^2\)
\(\Rightarrow-2\sqrt{25-9x^2}=a^2-10\)
\(\Rightarrow2\sqrt{25-9x^2}=10-a^2\)
\(\Rightarrow10+2\sqrt{25-9x^2}=20-a^2\)
\(\Rightarrow P=\dfrac{\sqrt{10+2\sqrt{25-9x^2}}}{x}=\dfrac{\sqrt{20-a^2}}{x}\)
\(\sqrt{5+3x}-\sqrt{5-3x}=a\\ \Rightarrow\left(\sqrt{5+3x}-\sqrt{5-3x}\right)^2=a^2\\ \Rightarrow5+3x-2\sqrt{\left(5+3x\right)\left(5-3x\right)}+5-3x=a^2\\ \Rightarrow2\sqrt{25-9x^2}=10-a^2\\ \Rightarrow4\left(25-9x^2\right)=\left(10-a^2\right)^2\\ \Rightarrow100-36x^2=100-20a^2+a^4\\ \Rightarrow36x^2=20a^2-a^4\\ \Rightarrow x^2=\dfrac{20a^2-a^4}{36}\\ \Rightarrow x=\dfrac{\sqrt{a^2\left(20-a^2\right)}}{6}\)
\(\Rightarrow P=\dfrac{\sqrt{10+2\sqrt{25-9x^2}}}{x}\\ =\dfrac{\sqrt{10+10-a^2}}{\dfrac{\sqrt{a^2\left(20-a^2\right)}}{6}}=6\sqrt{\dfrac{20-a^2}{a^2\left(20-a^2\right)}}=\dfrac{6}{\left|a\right|}\)
ai nay dung kinh nghiem la chinh
cau a)
ta thay \(10+6\sqrt{3}=\left(1+\sqrt{3}\right)^3\)
\(6+2\sqrt{5}=\left(1+\sqrt{5}\right)^2\)
khi do \(x=\frac{\sqrt[3]{\left(\sqrt{3}+1\right)^3}\left(\sqrt{3}-1\right)}{\sqrt{\left(1+\sqrt{5}\right)^2}-\sqrt{5}}\)
\(x=\frac{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}{1+\sqrt{5}-\sqrt{5}}\)
\(x=\frac{3-1}{1}=2\)
suy ra
x^3-4x+1=1
A=1^2018
A=1
b)
ta thay
\(7+5\sqrt{2}=\left(1+\sqrt{2}\right)^3\)
khi do
\(x=\sqrt[3]{\left(1+\sqrt{2}\right)^3}-\frac{1}{\sqrt[3]{\left(1+\sqrt{2}\right)^3}}\)
\(x=1+\sqrt{2}-\frac{1}{1+\sqrt{2}}=\frac{\left(1+\sqrt{2}\right)^2-1}{1+\sqrt{2}}=\frac{2+2\sqrt{2}}{1+\sqrt{2}}\)
x=2
thay vao
x^3+3x-14=0
B=0^2018
B=0
TL:
\(A=\frac{\sqrt{x+2}}{\sqrt{x-5}}\) mà x = 9
\(A=\frac{\sqrt{0+2}}{\sqrt{9-2}}\)
\(A=\frac{\sqrt{11}}{2}\)
b) chưa bt làm
Lời giải:
a) \(A=4\sqrt{x}-\frac{(\sqrt{x}+3)^2(\sqrt{x}-3)}{x-9}=4\sqrt{x}-\frac{(\sqrt{x}+3)(x-9)}{x-9}=4\sqrt{x}-(\sqrt{x}+3)\)
\(=3\sqrt{x}-3\)
b)
\(B=\frac{\sqrt{9x^2+12x+4}}{3x+2}=\frac{\sqrt{(3x)^2+2.3x.2+2^2}}{3x+2}=\frac{\sqrt{(3x+2)^2}}{3x+2}=\frac{|3x+2|}{3x+2}\)
\(B=1\) nếu $x>\frac{-2}{3}$
$B=-1$ nếu $x< \frac{-2}{3}$
b) Ta có: \(x+\sqrt{3}=2\Leftrightarrow x-2=-\sqrt{3}\Leftrightarrow\left(x-2\right)^2=3\Leftrightarrow x^2-4x+1=0\)
\(B=x^5-3x^4-3x^3+6x^2-20x+2021\)
\(B=\left(x^5-4x^4+x^3\right)+\left(x^4-4x^3+x^2\right)+\left(5x^2-20x+5\right)+2016\)
\(B=x^3\left(x^2-4x+1\right)+x^2\left(x^2-4x+1\right)+5\left(x^2-4x+1\right)+2016\)
Thế \(x^2-4x+1=0\)\(\Rightarrow B=2016.\)
P=\(\frac{\sqrt{10+2\sqrt{25-9x^2}}}{x}\)
P=\(\frac{\sqrt{10+2\sqrt{\left(5+3x\right)\left(5-3x\right)}}}{x}\)
P=\(\frac{\sqrt{10+10-a^2}}{x}\)(Vì a2=\(\left(\sqrt{5+3x}-\sqrt{5-3x}\right)^2\)=10-2\(\sqrt{\left(5+3x\right)\left(5-3x\right)}\))
\(\sqrt{5+3x}-\sqrt{5-3x}=a\)
\(\Leftrightarrow\left(\sqrt{5+3x}-\sqrt{5-3x}\right)^2=a^2\)
\(\Leftrightarrow5+3x+5-3x-2\sqrt{\left(5+3x\right)\left(5-3x\right)}=a^2\)
\(\Leftrightarrow10-2\sqrt{\left(5+3x\right)\left(5-3x\right)}=a^2\)
\(\Leftrightarrow2\sqrt{\left(5+3x\right)\left(5-3x\right)}=10-a^2\)
Thế vào P ta được:
\(P=\frac{\sqrt{10+2\sqrt{25-9x^2}}}{x}=\frac{\sqrt{10+2\sqrt{\left(5-3x\right)\left(5+3x\right)}}}{x}\)
\(=\frac{\sqrt{10+10-a^2}}{x}\)
\(=\frac{\sqrt{20-a^2}}{x}\)
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