\(\sqrt{x+9}=7\)

b) \(4\sqrt{2x+3}-\sqr...">

K
Khách

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AH
Akai Haruma
Giáo viên
31 tháng 7 2021

Lời giải:

a. ĐKXĐ: $x\geq -9$

PT $\Leftrightarrow x+9=7^2=49$

$\Leftrightarrow x=40$ (tm)

b. ĐKXĐ: $x\geq \frac{-3}{2}$

PT $\Leftrightarrow 4\sqrt{2x+3}-\sqrt{4(2x+3)}+\frac{1}{3}\sqrt{9(2x+3)}=15$

$\Leftrightarrow 4\sqrt{2x+3}-2\sqrt{2x+3}+\sqrt{2x+3}=15$

$\Leftrgihtarrow 3\sqrt{2x+3}=15$

$\Leftrightarrow \sqrt{2x+3}=5$

$\Leftrightarrow 2x+3=25$

$\Leftrightarrow x=11$ (tm)

 

AH
Akai Haruma
Giáo viên
31 tháng 7 2021

c.

PT \(\Leftrightarrow \left\{\begin{matrix} 2x+1\geq 0\\ x^2-6x+9=(2x+1)^2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\geq \frac{-1}{2}\\ 3x^2+10x-8=0\end{matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} x\geq \frac{-1}{2}\\ (3x-2)(x+4)=0\end{matrix}\right.\)

\(\Leftrightarrow x=\frac{2}{3}\)

d. ĐKXĐ: $x\geq 1$

PT \(\Leftrightarrow \sqrt{(x-1)+4\sqrt{x-1}+4}-\sqrt{(x-1)+6\sqrt{x-1}+9}=9\)

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

\(\Leftrightarrow \sqrt{x-1}+2-(\sqrt{x-1}+3)=9\)

\(\Leftrightarrow -1=9\) (vô lý)

Vậy pt vô nghiệm.

 

AH
Akai Haruma
Giáo viên
23 tháng 10 2020

Lời giải:

a) ĐK: $x\geq 2$

PT $\Leftrightarrow \sqrt{(x-2)(x+2)}-3\sqrt{x-2}=0$

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

\(\Rightarrow \left[\begin{matrix} \sqrt{x-2}=0\\ \sqrt{x+2}-3=0\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=2\\ x=7\end{matrix}\right.\) (thỏa mãn)

Vậy..........

b) ĐK: $x\geq 0$

PT $\Leftrightarrow (\sqrt{x}-3)^2=0$

$\Leftrightarrow \sqrt{x}-3=0$

$\Leftrightarrow x=9$ (thỏa mãn)

c) ĐK: $x\geq 3$

PT $\Leftrightarrow \sqrt{9(x-3)}+\sqrt{x-3}-\frac{1}{2}\sqrt{4(x-3)}=7$

$\Leftrightarrow 3\sqrt{x-3}+\sqrt{x-3}-\sqrt{x-3}=7$

$\Leftrightarrow 3\sqrt{x-3}=7$

$\Leftrightarrow x-3=(\frac{7}{3})^2$

$\Rightarrow x=\frac{76}{9}$

d)

ĐK: $x\geq \frac{-1}{2}$

PT $\Leftrightarrow 3\sqrt{4(2x+1)}-\frac{1}{3}\sqrt{9(2x+1)}-\frac{1}{2}\sqrt{25(2x+1)}+\sqrt{\frac{1}{4}(2x+1)}=6$

$\Leftrightarrow 6\sqrt{2x+1}-\sqrt{2x+1}-\frac{5}{2}\sqrt{2x+1}+\frac{1}{2}\sqrt{2x+1}=6$

$\Leftrightarrow 3\sqrt{2x+1}=6$

$\Leftrightarrow \sqrt{2x+1}=2$

$\Rightarrow x=\frac{3}{2}$ (thỏa mãn)

23 tháng 10 2020

cảm ơn nha <3

AH
Akai Haruma
Giáo viên
14 tháng 10 2018

1)

ĐK: \(x\geq 5\)

PT \(\Leftrightarrow \sqrt{4(x-5)}+3\sqrt{\frac{x-5}{9}}-\frac{1}{3}\sqrt{9(x-5)}=6\)

\(\Leftrightarrow \sqrt{4}.\sqrt{x-5}+3\sqrt{\frac{1}{9}}.\sqrt{x-5}-\frac{1}{3}.\sqrt{9}.\sqrt{x-5}=6\)

\(\Leftrightarrow 2\sqrt{x-5}+\sqrt{x-5}-\sqrt{x-5}=6\)

\(\Leftrightarrow 2\sqrt{x-5}=6\Rightarrow \sqrt{x-5}=3\Rightarrow x=3^2+5=14\)

AH
Akai Haruma
Giáo viên
14 tháng 10 2018

2)

ĐK: \(x\geq -1\)

\(\sqrt{x+1}+\sqrt{x+6}=5\)

\(\Leftrightarrow (\sqrt{x+1}-2)+(\sqrt{x+6}-3)=0\)

\(\Leftrightarrow \frac{x+1-2^2}{\sqrt{x+1}+2}+\frac{x+6-3^2}{\sqrt{x+6}+3}=0\)

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

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

\(\frac{1}{\sqrt{x+1}+2}+\frac{1}{\sqrt{x+6}+3}>0, \forall x\geq -1\) nên $x-3=0$

\(\Rightarrow x=3\) (thỏa mãn)

Vậy .............

Giải các phương trình sau: 1. a. \(\sqrt{x+3}-\sqrt{x-4}=1\) b. \(\sqrt{10-x}+\sqrt{x+3}=5\) c. \(\sqrt{15-x}+\sqrt{3-x}=6\) d. \(\sqrt{x-1}+\sqrt{x+1}=2\) e. \(\sqrt{4x+1}-\sqrt{3x+4}=1\) f. \(\sqrt{x-2\sqrt{x-1}}-\sqrt{x-1}=1\) g. \(\sqrt{x+\sqrt{2x+1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\) h. \(\sqrt{x+\sqrt{6x-9}}+\sqrt{x-\sqrt{6x-9}}=\sqrt{6}\) i. \(\sqrt{x^2-4x+4}+\sqrt{x^2-6x+9}=1\) k. \(\sqrt{x+4-4\sqrt{x}}+\sqrt{x+9-6\sqrt{x}}=1\) l....
Đọc tiếp

Giải các phương trình sau:

1.

a. \(\sqrt{x+3}-\sqrt{x-4}=1\)

b. \(\sqrt{10-x}+\sqrt{x+3}=5\)

c. \(\sqrt{15-x}+\sqrt{3-x}=6\)

d. \(\sqrt{x-1}+\sqrt{x+1}=2\)

e. \(\sqrt{4x+1}-\sqrt{3x+4}=1\)

f. \(\sqrt{x-2\sqrt{x-1}}-\sqrt{x-1}=1\)

g. \(\sqrt{x+\sqrt{2x+1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\)

h. \(\sqrt{x+\sqrt{6x-9}}+\sqrt{x-\sqrt{6x-9}}=\sqrt{6}\)

i. \(\sqrt{x^2-4x+4}+\sqrt{x^2-6x+9}=1\)

k. \(\sqrt{x+4-4\sqrt{x}}+\sqrt{x+9-6\sqrt{x}}=1\)

l. \(\sqrt{x+6-4\sqrt{x+2}}+\sqrt{x+11-6\sqrt{x+2}}=1\)

m. \(\sqrt{x+2-4\sqrt{x-2}}+\sqrt{x+7-6\sqrt{x-2}=1}\)

n. \(\sqrt{x}+\sqrt{x+\sqrt{1-x}}=1\)

o. \(\sqrt{1-\sqrt{x^2-x}}=\sqrt{x}-1\)

p. \(\sqrt{x^2+6}=x-2\sqrt{x^2-1}\)

q. \(\sqrt{2x^2+8x+6}+\sqrt{x^2-1}=2x+2\)

r. \(\sqrt{x-7}+\sqrt{9-x}=x^2-16x+66\)

s. \(\sqrt{2x-1}+\sqrt{x-2}=\sqrt{x+1}\)

t. \(\sqrt{3x+15}-\sqrt{4x-17}=\sqrt{x+2}\)

u. \(\sqrt{x-1}+\sqrt{x+3}+2\sqrt{\left(x-1\right)\left(x^2-3x+5\right)}=4-2x\)

v. \(\sqrt{x+1}+\sqrt{x+10}=\sqrt{x+2}+\sqrt{x+5}\)

w. \(\sqrt{2x+3+\sqrt{x+2}}+\sqrt{2x+2-\sqrt{x+2}}=1+2\sqrt{x+2}\)

x. \(\sqrt{2x^2-9x+4}+3\sqrt{2x-1}=\sqrt{2x^2+21x-11}\)

y. \(\sqrt{1-x}+\sqrt{x^2-3x+2}+\left(x-2\right)\sqrt{\dfrac{x-1}{x-2}}=3\)

z. \(\left(x-2\right)\left(x+2\right)+4\left(x-2\right)\sqrt{\dfrac{x+2}{x-2}}=-3\)

2.

a. \(\dfrac{2+\sqrt{x}}{\sqrt{2}+\sqrt{2+\sqrt{x}}}+\dfrac{2-\sqrt{x}}{\sqrt{2}-\sqrt{2-\sqrt{x}}}=\sqrt{2}\)

b. \(\dfrac{x}{2+\dfrac{x}{2+\dfrac{x}{2+\dfrac{...}{2+\dfrac{x}{1+\sqrt{1+x}}}}}}=8\) (vế trái có 100 dấu phân thức)

c. \(\sqrt[3]{x+1}+\sqrt[3]{7-x}=2\)

d. \(\sqrt[4]{1-x}+\sqrt[4]{2-x}=\sqrt[4]{3-2x}\)

e. \(\sqrt[4]{1-x^2}+\sqrt[4]{1+x}+\sqrt[4]{1-x}=3\)

f. \(\dfrac{\sqrt[3]{7-x}-\sqrt[3]{x-5}}{\sqrt[3]{7-x}+\sqrt[3]{x-5}}=6-x\)

g. \(\sqrt[3]{x+1}+\sqrt[3]{x+2}+\sqrt[3]{x+3}=0\)

h. \(\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{\left(x-1\right)^2}+\sqrt[3]{x^2-1}=1\)

i. \(\sqrt[3]{x+1}+\sqrt[3]{x-1}=\sqrt[3]{5x}\)

k. \(\sqrt[3]{x-2}+\sqrt{x+1}=3\)

l. \(\sqrt[3]{24+x}+\sqrt{12-x}=6\)

m. \(\sqrt[3]{2-x}+\sqrt{x-1}=1\)

n. \(1+\sqrt[3]{x-16}=\sqrt[3]{x+3}\)

o. \(\sqrt[3]{25+x}+\sqrt[3]{3-x}=4\)

p. \(\sqrt[3]{x+3}-\sqrt[3]{6-x}=1\)

Làm nhanh giúp mk nhé mn ơi

5
19 tháng 11 2018

Giải pt :

1

a. ĐKXĐ : \(x\ge4\)

Ta có :

\(\sqrt{x+3}-\sqrt{x-4}=1\\ \Leftrightarrow\sqrt{x+3}=1+\sqrt{x-4}\\ \Leftrightarrow x+3=x-3+2\sqrt{x-4}\\ \Leftrightarrow6=2\sqrt{x-4}\)

\(\Leftrightarrow3=\sqrt{x-4}\\ \Leftrightarrow x-4=9\)

\(\Leftrightarrow x=13\) (TM ĐKXĐ)

Vậy \(S=\left\{13\right\}\)

b.ĐKXĐ : \(-3\le x\le10\)

Ta có :

\(\sqrt{10-x}+\sqrt{x+3}=5\\ \Leftrightarrow13+2\sqrt{-x^2+7x+30}=25\\ \Leftrightarrow\sqrt{-x^2+7x+30}=6\\ \Leftrightarrow-x^2+7x+30=36\\ \Leftrightarrow-x^2+7x-6=0\\ \Leftrightarrow-x^2+x+6x-6=0\\ \Leftrightarrow-x\left(x-1\right)+6\left(x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left(6-x\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\left(TMĐKXĐ\right)\\x=6\left(TMĐKXĐ\right)\end{matrix}\right.\)

Vậy \(S=\left\{1;6\right\}\)

19 tháng 11 2018

Câu c,d làm giống câu b

Câu e làm giống câu a

NV
13 tháng 8 2020

5.

\(\Leftrightarrow x^2+7-\left(x+4\right)\sqrt{x^2+7}+4x=0\)

Đặt \(\sqrt{x^2+7}=t>0\)

\(\Rightarrow t^2-\left(x+4\right)t+4x=0\)

\(\Delta=\left(x+4\right)^2-16x=\left(x-4\right)^2\)

\(\Rightarrow\left[{}\begin{matrix}t=\frac{x+4+x-4}{2}=x\\t=\frac{x+4-x+4}{2}=4\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt{x^2+7}=x\left(x\ge0\right)\\\sqrt{x^2+7}=4\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+7=x^2\left(vn\right)\\x^2+7=16\end{matrix}\right.\)

Câu 6 bạn coi lại đề

NV
13 tháng 8 2020

4.

ĐKXĐ: ...

Đặt \(\sqrt{x+3}=a\ge0\)

\(\Rightarrow x+a=\sqrt{5x^2-a^2}\)

\(\Rightarrow x^2+2ax+a^2=5x^2-a^2\)

\(\Rightarrow2x^2-ax-a^2=0\)

\(\Rightarrow\left(x-a\right)\left(2x+a\right)=0\)

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

\(\Rightarrow\left[{}\begin{matrix}\sqrt{x+3}=x\left(x\ge0\right)\\\sqrt{x+3}=-2x\left(x\le0\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x+3=x^2\left(x\ge0\right)\\x+3=4x^2\left(x\le0\right)\end{matrix}\right.\)

 

14 tháng 8 2017

I) xd mọi x

\(\sqrt{x^2-8x+16}+\sqrt{x^2-10x+25}=9\)

\(\sqrt{\left(x-4\right)^2}+\sqrt{\left(x-5\right)^2}=9=>\left|x-4\right|+\left|x-5\right|=9\)

\(\left[{}\begin{matrix}x< 4\Rightarrow4-x+5-x=>x=0\left(n\right)\\4\le x< 5\Rightarrow x-4+5-x=9\left(vn\right)\\x\ge5\Rightarrow x-4+x-5=9\Rightarrow x=9\left(n\right)\\\end{matrix}\right.\)

kết luận

\(\left[{}\begin{matrix}x=0\\x=9\end{matrix}\right.\)

28 tháng 1 2019

Em xin phép làm bài EZ nhất :)

4,ĐK :\(\forall x\in R\)

Đặt \(x^2+x+2=t\) (\(t\ge\dfrac{7}{4}\))

\(PT\Leftrightarrow\sqrt{t+5}+\sqrt{t}=\sqrt{3t+13}\)

\(\Leftrightarrow2t+5+2\sqrt{t\left(t+5\right)}=3t+13\)

\(\Leftrightarrow t+8=2\sqrt{t^2+5t}\)

\(\Leftrightarrow\left\{{}\begin{matrix}t\ge-8\\\left(t+8\right)^2=4t^2+20t\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}t\ge\dfrac{7}{4}\\3t^2+4t-64=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}t\ge\dfrac{7}{4}\\\left(t-4\right)\left(3t+16\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}t\ge\dfrac{7}{4}\\\left[{}\begin{matrix}t=4\left(tm\right)\\t=-\dfrac{16}{3}\left(l\right)\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow x^2+x+2=4\)\(\Leftrightarrow x^2+x-2=0\)

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

Vậy ....

20 tháng 8 2019

a) \(\sqrt{4x}=10\) (ĐKXĐ: 4x>=0 <=> x>=0)

\(\Leftrightarrow4x=100\)

\(\Leftrightarrow x=25\)

\(S=\left\{25\right\}\)

b) \(\sqrt{x^2-2x+1}=8\)

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

\(\Leftrightarrow x-1=8\)

\(\Leftrightarrow x=9\)

\(S=\left\{9\right\}\)

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

\(\Leftrightarrow\sqrt{\left(x-3\right)^2}=\sqrt{\left(1-3x\right)^2}\)

\(\Leftrightarrow x-3=1-3x\) hoặc \(\Leftrightarrow x-3=-1+3x\)

\(\Leftrightarrow x+3x=1+3\) \(\Leftrightarrow x-3x=-1+3\)

\(\Leftrightarrow4x=4\) \(\Leftrightarrow-2x=2\)

\(\Leftrightarrow x=1\) \(\Leftrightarrow x=-1\)

\(S=\left\{1;-1\right\}\)

d) \(\sqrt{2x-5}=x-2\)

\(\Leftrightarrow2x-5=x^2-4x+4\)

\(\Leftrightarrow-x^2+2x+4x-5-4=0\)

\(\Leftrightarrow-x^2+6x-9=0\)

\(\Leftrightarrow x^2-6x+9=0\)

\(\Leftrightarrow\left(x-3\right)^2=0\)

\(\Leftrightarrow x-3=0\)

\(\Leftrightarrow x=3\)

\(S=\left\{3\right\}\)

e) \(\sqrt{x^2-2x+1}=\sqrt{x+1}\)

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

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

\(\Leftrightarrow x^2-3x=0\)

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

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

\(\Leftrightarrow x=3\)

\(S=\left\{0;3\right\}\)

g) \(\sqrt{x^2-9}-\sqrt{x-3}=0\) ( ĐKXĐ: x-3>=0 <=> x>=3)

\(\Leftrightarrow\sqrt{x^2-9}=\sqrt{x-3}\)

\(\Leftrightarrow x^2-9=x-3\)

\(\Leftrightarrow x^2-x-6=0\)

\(\Leftrightarrow x^2-3x+2x-6=0\)

\(\Leftrightarrow\left(x^2+2x\right)-\left(3x+6\right)=0\)

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

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

\(\Leftrightarrow x+2=0\) hoặc \(\Leftrightarrow x-3=0\)

\(\Leftrightarrow x=-2\) \(\Leftrightarrow x=3\)

\(S=\left\{-2;3\right\}\)

h) \(\sqrt{x^2-4x+4}+\sqrt{x^2-6x+9}=1\)

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

\(\Leftrightarrow x-2+x-3-1=0\)

\(\Leftrightarrow2x-6=0\)

\(\Leftrightarrow x=3\)

\(S=\left\{3\right\}\)

i) \(\sqrt{\frac{2x-3}{x-1}}=2\)

\(\Leftrightarrow\frac{2x-3}{x-1}=4\)

\(\Leftrightarrow4\left(x-1\right)=2x-3\)

\(\Leftrightarrow4x-4-2x+3=0\)

\(\Leftrightarrow2x-1=0\)

\(\Leftrightarrow x=\frac{1}{2}\)

\(S=\left\{\frac{1}{2}\right\}\)

l) \(x+y+12=4\sqrt{x}+6\sqrt{y-1}\)

\(\Leftrightarrow x+y-4\sqrt{x}+12-6\sqrt{y-1}=0\)

\(\Leftrightarrow\left(x-4\sqrt{x}+4\right)+\left(y-1-6\sqrt{y-1}+9\right)=0\)

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

\(\Leftrightarrow\sqrt{x}-2=0\) hoặc \(\Leftrightarrow\sqrt{y-1}-3=0\)

\(\Leftrightarrow\sqrt{x}=2\) \(\Leftrightarrow\sqrt{y-1}=3\)

\(\Leftrightarrow x=4\) \(\Leftrightarrow y-1=9\)

\(\Leftrightarrow y=10\)

KẾT luận : ..............

Tới đây nhé, nếu mai chưa ai giải thì mình giải hộ cho

CHÚC BẠN HỌC TỐT!

21 tháng 8 2019

m) \(\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8+6\sqrt{x-1}}=5\)

<=> \(\sqrt{\left(x-1\right)-4\sqrt{x-1}+4}+\sqrt{\left(x-1\right)+6\sqrt{x-1}+9}=5\)

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

<=>\(\sqrt{x-1}+2+\sqrt{x-1}+3=5\)

<=> \(2\sqrt{x-1}=0\)

<=> \(\sqrt{x-1}=0\) <=>x=1

Vậy \(S=\left\{1\right\}\)

n) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\) (*) ( đk \(x\ge\frac{1}{2}\))

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

<=> \(x+\sqrt{2x-1}+x-\sqrt{2x-1}+2\sqrt{x^2-2x+1}=2\)

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

<=> x+\(\left|x-1\right|=2\)(1)

TH1: \(\frac{1}{2}\le x\le1\)

Từ (1) => x+1-x=2

<=> 1=2(vô lý)

TH2: x>1

Từ (1)=> x+x-1=2

<=> 2x=3<=> \(x=\frac{2}{3}\)(tm pt (*))

Vậy \(S=\left\{\frac{2}{3}\right\}\)

p) \(\sqrt{2x-1}+\sqrt{x-2}=\sqrt{x+1}\) (*) (đk :\(x\ge2\))

Đặt \(\left\{{}\begin{matrix}x-2=a\left(a\ge0\right)\\x+1=b\left(b\ge0\right)\end{matrix}\right.\) =>a+b=2x-1

\(\sqrt{a+b}+\sqrt{a}=\sqrt{b}\)

<=> \(\sqrt{a+b}=\sqrt{b}-\sqrt{a}\)

<=> \(a+b=b-2\sqrt{ab}+a\)

<=> 0=\(-2\sqrt{ab}\)

=> \(\left[{}\begin{matrix}a=0\\b=0\end{matrix}\right.\) <=> \(\left[{}\begin{matrix}x+1=0\\x-2=0\end{matrix}\right.\) <=> \(\left[{}\begin{matrix}x=-1\\x=2\end{matrix}\right.\) => x=2 (vì x=-1 không thỏa mãn pt(*))

Vậy \(S=\left\{2\right\}\)

q) \(\sqrt{x-7}+\sqrt{9-x}=x^2-16x+66\)(*) (đk : \(7\le x\le9\))

Với a,b\(\ge0\) có: \(\sqrt{a}+\sqrt{b}\le2\sqrt{\frac{a+b}{2}}\)(tự cm nha) .Dấu "=" xảy ra <=> a=b

Áp dụng bđt trên có:

\(\sqrt{x-7}+\sqrt{9-x}\le2\sqrt{\frac{x-7+9-x}{2}}=2\sqrt{\frac{2}{2}}=2\) (1)

Có x2-16x+66=(x2-16x+64)+2=(x-8)2+2 \(\ge2\) với mọi x (2)

Từ (1),(2) .Dấu "=" xảy ra <=> \(\left\{{}\begin{matrix}x-7=9-x\\x-8=0\end{matrix}\right.\)<=>\(\left\{{}\begin{matrix}2x=16\\x=8\end{matrix}\right.\)<=>\(\left\{{}\begin{matrix}x=8\\x=8\end{matrix}\right.\)<=> x=8( tm pt (*))

Vậy \(S=\left\{8\right\}\)

AH
Akai Haruma
Giáo viên
23 tháng 11 2018

Câu a:

ĐKXĐ:...........

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

\(\Rightarrow \left\{\begin{matrix} 2x+1\geq 0\\ x^2-x+9=(2x+1)^2=4x^2+4x+1\end{matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} x\geq \frac{-1}{2}\\ 3x^2+5x-8=0\end{matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} x\geq \frac{-1}{2}\\ 3x(x-1)+8(x-1)=0\end{matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} x\geq \frac{-1}{2}\\ (x-1)(3x+8)=0\end{matrix}\right.\Rightarrow x=1\)

Vậy.....

AH
Akai Haruma
Giáo viên
23 tháng 11 2018

Câu b:

ĐKXĐ:.........

Ta có: \(\sqrt{5x+7}-\sqrt{x+3}=\sqrt{3x+1}\)

\(\Rightarrow (\sqrt{5x+7}-\sqrt{x+3})^2=3x+1\)

\(\Leftrightarrow 5x+7+x+3-2\sqrt{(5x+7)(x+3)}=3x+1\)

\(\Leftrightarrow 3(x+3)=2\sqrt{(5x+7)(x+3)}\)

\(\Leftrightarrow \sqrt{x+3}(3\sqrt{x+3}-2\sqrt{5x+7})=0\)

\(x\geq -\frac{7}{5}\Rightarrow \sqrt{x+3}>0\). Do đó:

\(3\sqrt{x+3}-2\sqrt{5x+7}=0\)

\(\Rightarrow 9(x+3)=4(5x+7)\)

\(\Rightarrow 11x=-1\Rightarrow x=\frac{-1}{11}\) (thỏa mãn)

Vậy..........

NV
11 tháng 8 2020

5.

ĐKXĐ: ...

\(\Leftrightarrow3x^2-14x-5+\sqrt{3x+1}-4+1-\sqrt{6-x}=0\)

\(\Leftrightarrow\left(3x+1\right)\left(x-5\right)+\frac{3\left(x-5\right)}{\sqrt{3x+1}+4}+\frac{x-5}{1+\sqrt{6-x}}=0\)

\(\Leftrightarrow\left(x-5\right)\left(3x+1+\frac{3}{\sqrt{3x+1}+4}+\frac{1}{1+\sqrt{6-x}}\right)=0\)

\(\Leftrightarrow x=5\)

6.

ĐKXĐ: \(-4\le x\le4\)

\(\Leftrightarrow\frac{\left(\sqrt{x+4}-2\right)\left(\sqrt{x+4}+2\right)\left(\sqrt{4-x}+2\right)}{\sqrt{x+4}+2}=2x\)

\(\Leftrightarrow\frac{x\left(\sqrt{4-x}+2\right)}{\sqrt{x+4}+2}=2x\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\\frac{\sqrt{4-x}+2}{\sqrt{x+4}+2}=2\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow\sqrt{4-x}+2=2\sqrt{x+4}+4\)

\(\Leftrightarrow2\sqrt{x+4}-\frac{4}{5}+\frac{14}{5}-\sqrt{4-x}=0\)

\(\Leftrightarrow\frac{2\left(x+4-\frac{4}{25}\right)}{\sqrt{x+4}+\frac{2}{5}}+\frac{\frac{196}{25}-4+x}{\frac{14}{5}+\sqrt{4-x}}=0\)

\(\Leftrightarrow\left(x-\frac{96}{25}\right)\left(\frac{2}{\sqrt{x+4}+\frac{2}{5}}+\frac{1}{\frac{14}{5}+\sqrt{4-x}}\right)=0\)

\(\Rightarrow x=\frac{96}{25}\)

NV
11 tháng 8 2020

1.

Bạn coi lại đề

2.

ĐKXĐ: \(1\le x\le2\)

Nhận thấy \(\sqrt{x+2}+\sqrt{x-1}>0;\forall x\) , nhân 2 vế của pt với nó:

\(\left(\sqrt{x+2}+\sqrt{x-1}\right)\left(\sqrt{x+2}-\sqrt{x-1}\right)\left(\sqrt{2-x}+1\right)=\sqrt{x+2}+\sqrt{x-1}\)

\(\Leftrightarrow3\left(\sqrt{2-x}+1\right)=\sqrt{x+2}+\sqrt{x-1}\)

\(\Leftrightarrow3\sqrt{2-x}+3=\sqrt{x+2}+\sqrt{x-1}\)

\(\Leftrightarrow3\sqrt{2-x}+2-\sqrt{x+2}+1-\sqrt{x-1}=0\)

\(\Leftrightarrow3\sqrt{2-x}+\frac{2-x}{2+\sqrt{x+2}}+\frac{2-x}{1+\sqrt{x-1}}=0\)

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

\(\Leftrightarrow\sqrt{2-x}=0\Rightarrow x=2\)