\(\sqrt{25x^2+80x+64}+\sqrt{9x^2-6x+1}=\sqrt{4x^2+36x+81}\)

\(pt\Leftrightarrow\sqrt{\left(5x+8\right)^2}+\sqrt{\left(3x-1\right)^2}=\sqrt{\left(2x+9\right)^2}\)

\(\Leftrightarrow\left|5x+8\right|+\left|3x-1\right|=\left|2x+9\right|\)

Áp dụng BĐT \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:

\(VT=\left|5x+8\right|+\left|-\left(3x-1\right)\right|\)

\(=\left|5x+8\right|+\left|-3x+1\right|\)

\(\ge\left|5x+8-3x+1\right|=\left|2x+9\right|=VP\)

Đẳng thức xảy ra khi \(-\frac{8}{5}\le x\le\frac{1}{3}\)

P.s:thực ra thì áp dụng căn a+căn b>= căn a+b ngay từ đầu luôn cx dc tùy

Lời giải:

Đặt \(\sqrt{6x-1}=a;\sqrt{9x^2-1}=b\). Khi đó :

\(6x-9x^2=a^2-b^2\)

PT tương đương:

\(a+b=a^2-b^2\)

\(\Leftrightarrow (a+b)[1-(a-b)]=0\)

\(\Leftrightarrow \) \(\left[{}\begin{matrix}a+b=0\\a-b=1\end{matrix}\right.\)

+) Nếu \(a+b=0\Leftrightarrow \sqrt {6x-1}+\sqrt{9x^2-1}=0\)

Vì \(\sqrt{6x-1}\geq 0; \sqrt{9x^2-1}\geq 0\) nên điều trên xảy ra khi mà

\(\sqrt{6x-1}=\sqrt{9x^2-1}=0\) (vô lý)

+) Nếu \(a-b=1\Leftrightarrow \sqrt{6x-1}-\sqrt{9x^2-1}=1\)

\(\Leftrightarrow \sqrt{6x-1}=\sqrt{9x^2-1}+1\)

\(\Leftrightarrow 6x-1=9x^2-1+1+2\sqrt{9x^2-1}\)

\(\Leftrightarrow 9x^2-6x+1+2\sqrt{9x^2-1}=0\)

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

Vì \((3x-1)^2\geq 0; \sqrt{(3x-1)(3x+1)}\geq 0\) nên điều trên xảy ra khi mà:

\((3x-1)^2=\sqrt{(3x-1)(3x+1)}=0\Leftrightarrow x=\frac{1}{3}\)

Thử lại thấy đúng.

Vậy \(x=\frac{1}{3}\)

a.

\(\sqrt{4x^2+4x+1}-\sqrt{25x^2+10x+1}=0\)

\(\Leftrightarrow\sqrt{\left(2x+1\right)^2}-\sqrt{\left(5x+1\right)^2}=0\)

\(\Leftrightarrow2x+1-\left(5x+1\right)=0\)

\(\Leftrightarrow-3x=0\Leftrightarrow x=0\)

b.

\(\sqrt{x^4-16x^2+64}=\sqrt{25x^2+10x+1}\)

\(\Leftrightarrow\sqrt{\left(x^2-8\right)^2}=\sqrt{\left(5x+1\right)^2}\)

\(\Leftrightarrow x^2-8=5x+1\)

\(\Leftrightarrow x^2-5x+\dfrac{25}{4}=\dfrac{61}{4}\)

\(\Leftrightarrow\left(x-\dfrac{5}{2}\right)^2=\dfrac{61}{4}\)

............................

tương tự ..

c: \(\Leftrightarrow\sqrt{x-5}\left(\sqrt{x+5}-1\right)=0\)

=>x-5=0 hoặc x+5=1

=>x=-4 hoặc x=5

d: \(\Leftrightarrow\sqrt{2x+3}\left(\sqrt{2x-3}-2\right)=0\)

=>2x+3=0 hoặc 2x-3=4

=>x=7/2 hoặc x=-3/2

e: \(\Leftrightarrow\sqrt{x-2}\left(1-3\sqrt{x+2}\right)=0\)

=>x-2=0 hoặc 3 căn x+2=1

=>x=2 hoặc x+2=1/9

=>x=-17/9 hoặc x=2

Điều kiện: 6x - 1 \(\ge\) 0 và 9x² - 1 \(\ge\) 0 

=> x \(\ge\) 1/6 và (3x -1).(3x+ 1) \(\ge\) 0 => x\(\ge\) 1/6 và 3x - 1\(\ge\) 0 => x\(\ge\)1/3

PT <=> \(\left(\sqrt{6x-1}-1\right)+\sqrt{\left(3x-1\right)\left(3x+1\right)}=0\)

<=> \(\frac{\left(\sqrt{6x-1}-1\right)\left(\sqrt{6x-1}+1\right)}{\sqrt{6x-1}+1}+\sqrt{\left(3x-1\right)\left(3x+1\right)}=0\)

<=> \(\frac{2.\left(3x-1\right)}{\sqrt{6x-1}+1}+\sqrt{\left(3x-1\right)}.\sqrt{3x+1}=0\)

<=> \(\left(\frac{2.\sqrt{3x-1}}{\sqrt{6x-1}+1}+\sqrt{3x+1}\right).\sqrt{3x-1}=0\)

<=> \(\frac{2.\sqrt{3x-1}}{\sqrt{6x-1}+1}+\sqrt{3x+1}=0\) hoặc \(\sqrt{3x-1}=0\)

+) \(\sqrt{3x-1}=0\) => x= 1/3 (thỏa mãn)

+) \(\frac{2.\sqrt{3x-1}}{\sqrt{6x-1}+1}+\sqrt{3x+1}=0\) Vô nghiệm Vì Với x \(\ge\) 1/3

=>  \(\frac{2.\sqrt{3x-1}}{\sqrt{6x-1}+1}+\sqrt{3x+1}\ge0+\sqrt{3.\frac{1}{3}+1}=\sqrt{2}>0\)

Vậy PT đã cho có 1 nghiệm là x = 1/3

đề sai r,,,,,,cái kia phải là x^2-x+1 chứ

nếu đúng như tôi thì bạn chỉ cần cho cái 2 vào trong căn rồi nhân liên hợp là ok

yes..thanks

Sửa đề: \(\sqrt{x-5}+\dfrac{1}{3}\sqrt{9x-45}=\dfrac{1}{5}\sqrt{25x-125}+6\)

\(\Leftrightarrow\sqrt{x-5}+\dfrac{1}{3}\cdot3\cdot\sqrt{x-5}-\dfrac{1}{5}\cdot5\sqrt{x-5}=6\)

\(\Leftrightarrow\sqrt{x-5}=6\)

=>x-5=36

hay x=41

a: =>2*căn x+5+căn x+5-1/3*3*căn x+5=4

=>2*căn(x+5)=4

=>căn (x+5)=2

=>x+5=4

=>x=-1

b: =>\(6\sqrt{x-1}-3\sqrt{x-1}-2\sqrt{x-1}+\sqrt{x-1}=16\)

=>2*căn x-1=16

=>x-1=64

=>x=65

c, \(\sqrt{\left(x-3\right)^2}-2\sqrt{\left(x-1\right)^2}+\sqrt{x^2}=0\\ \Leftrightarrow\left|x-3\right|-2\left|x-1\right|+\left|x\right|=0\left(1\right)\)

TH₁: \(x\ge3\)

\(\left(1\right)\Rightarrow x-3-2x+2+x=0\\ \Leftrightarrow-1=0\left(loại\right)\)

TH₂: \(2\le x< 3\)

\(\left(1\right)\Rightarrow3-x-2x+2+x=0\\ \Leftrightarrow-2x=-5\\ \Leftrightarrow x=\dfrac{5}{2}\left(tm\right)\)

TH₃: \(0\le x< 2\)

\(\left(1\right)\Rightarrow3-x+2x-2+x=0\\ \Leftrightarrow2x=1\\ \Leftrightarrow x=\dfrac{1}{2}\left(tm\right)\)

TH₄: \(x< 0\)

\(\left(1\right)\Rightarrow3-x+2x-2-x-=0\\ \Leftrightarrow1=0\left(loại\right)\)

Vậy \(x\in\left\{\dfrac{1}{2};\dfrac{5}{2}\right\}\)

a,\(\sqrt{\left(3x-1\right)^2}=5=>|3x-1|=5=>\left[{}\begin{matrix}3x-1=5\\3x-1=-5\end{matrix}\right.\)

\(=>\left[{}\begin{matrix}x=2\\x=-\dfrac{4}{3}\end{matrix}\right.\)

b, \(\sqrt{4x^2-4x+1}=3=\sqrt{\left(2x-1\right)^2}=3=>\left[{}\begin{matrix}2x-1=3\\2x-1=-3\end{matrix}\right.\)

\(=>\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\)

c, \(\sqrt{x^2-6x+9}+3x=4=>|x-3|=4-3x\)

TH1: \(|x-3|=x-3< =>x\ge3=>x-3=4-3x=>x=1,75\left(ktm\right)\)

TH2 \(|x-3|=3-x< =>x< 3=>3-x=4-3x=>x=0,5\left(tm\right)\)

Vậy x=0,5...

d, đk \(x\ge-1\)

=>pt đã cho \(< =>9\sqrt{x+1}-6\sqrt{x+1}+4\sqrt{x+1}=12\)

\(=>7\sqrt{x+1}=12=>x+1=\dfrac{144}{49}=>x=\dfrac{95}{49}\left(tm\right)\)

a) Ta có: \(\sqrt{\left(3x-1\right)^2}=5\)

\(\Leftrightarrow\left|3x-1\right|=5\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-1=5\\3x-1=-5\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}3x=6\\3x=-4\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-\dfrac{4}{3}\end{matrix}\right.\)

b) Ta có: \(\sqrt{4x^2-4x+1}=3\)

\(\Leftrightarrow\left|2x-1\right|=3\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-1=3\\2x-1=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}2x=4\\2x=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\)

c) Ta có: \(\sqrt{x^2-6x+9}+3x=4\)

\(\Leftrightarrow\left|x-3\right|=4-3x\)

\(\Leftrightarrow\left[{}\begin{matrix}x-3=4-23x\left(x\ge3\right)\\x-3=23x-4\left(x< 3\right)\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x+23x=4+3\\x-23x=4+3\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{7}{24}\left(loại\right)\\x=\dfrac{-4}{22}=\dfrac{-2}{11}\left(loại\right)\end{matrix}\right.\)