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Làm hơi tắt xíu, có gì ko hiểu cmt nha :>
\(a.\sqrt{x-1}=3\left(ĐK:x\ge1\right)\Leftrightarrow x-1=9\Leftrightarrow x=10\)
\(b.\sqrt{x^2-4x+4}=2\\ \Leftrightarrow\sqrt{\left(x-2\right)^2}=2\\ \Leftrightarrow\left|x-2\right|=2\\ \Leftrightarrow\left[{}\begin{matrix}x-2=2\left(x\ge2\right)\\2-x=2\left(x< 2\right)\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=4\\x=0\end{matrix}\right.\)
\(c.\sqrt{25x^2-10x+1}=4x-9\\ \Leftrightarrow\sqrt{\left(5x-1\right)^2}=4x-9\\ \Leftrightarrow\left|5x-1\right|=4x-9\\\Leftrightarrow \left[{}\begin{matrix}5x-1=4x-9\left(x\ge\frac{1}{5}\right)\\1-5x=4x-9\left(x< \frac{1}{5}\right)\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=-8\left(ktm\right)\\x=\frac{10}{9}\left(ktm\right)\end{matrix}\right.\)
\(d.\sqrt{x^2+2x+1}=\sqrt{x+1}\left(ĐK:x\ge-1\right)\\ \Leftrightarrow x^2+2x+1=x+1\\ \Leftrightarrow x^2+x=0\Leftrightarrow x\left(x+1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)
e. ĐK: \(\left[{}\begin{matrix}x\ge3\\x\le-3\end{matrix}\right.\)
\(\sqrt{x^2-9}+\sqrt{x^2-6x+9}=0\\ \Leftrightarrow\sqrt{\left(x-3\right)\left(x+3\right)}+\sqrt{\left(x-3\right)^2}=0\\ \Leftrightarrow\sqrt{x-3}\left(\sqrt{x+3}+\sqrt{x-3}\right)=0\\ \Leftrightarrow\sqrt{x-3}=0\\ \Leftrightarrow x-3=0\Leftrightarrow x=3\)
Câu cuối chưa nghĩ ra, sorry :<
d) Bài này có thể dùng hằng đẳng thức rồi phá dấu GTTĐ nhưng theo em là khá mất công nên bình phương lên rồi quy về pt bậc 2 cho lẹ:)
PT \(\Leftrightarrow4x^2-4x+1=x^2-6x+9\)
\(\Leftrightarrow3x^2+2x-8=0\Leftrightarrow\left[{}\begin{matrix}x=\frac{4}{3}\\x=-2\end{matrix}\right.\) (delta là ra:D)
Vậy..
a) \(\Leftrightarrow\sqrt{\left(x+3\right)^2}=4\)
\(\Leftrightarrow\left|x+3\right|=4\) \(\Leftrightarrow\left[{}\begin{matrix}x+3=4\\x+3=-4\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-7\end{matrix}\right.\) ( TM )
b) \(\Leftrightarrow\sqrt{\left(2x-1\right)^2}=5x+3\)
\(\Leftrightarrow\left|2x-1\right|=5x+3\)
\(\Leftrightarrow\left\{{}\begin{matrix}5x+3\ge0\\\left[{}\begin{matrix}2x-1=5x+3\\2x-1=-5x-3\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-\frac{3}{5}\\\left[{}\begin{matrix}x=-\frac{4}{3}\left(KTM\right)\\x=-\frac{2}{7}\left(TM\right)\end{matrix}\right.\end{matrix}\right.\)
a \(\sqrt{x^2+6x+9}=4\Leftrightarrow\sqrt{\left(x+3\right)^2=4}\)
\(\Leftrightarrow x+3=4\)
\(\Rightarrow x=1\)
a) \(\sqrt{x^2-6x+9}+x=11\)
\(\Rightarrow\sqrt{\left(x-3\right)^2}+x=11\)
\(\Rightarrow x-3+x=11\)
\(\Rightarrow2x=14\Rightarrow x=7\)
Vậy........
b) \(\sqrt{3x^2-4x+3}=1-2x\)
\(3x^2-4x+3=1-4x+4x^2\)
\(3x^2-4x^2-4x+4x=-2\)
\(-x^2=-2\)
\(2=x^2\Rightarrow\orbr{\begin{cases}x=\sqrt{2}\\x=-\sqrt{2}\end{cases}}\)
Vậy.........
d) \(\sqrt{4x^2-4x+1}=\sqrt{x^2-6x+9}\)
\(\Rightarrow2x-1=x-3\)
\(\Rightarrow x=1-3\)
\(\Rightarrow x=-2\)
Vậy x=-2
a) điều kiện xác định : \(x\ge1\)
ta có : \(\sqrt{\dfrac{x-1}{4}}-3=\sqrt{\dfrac{4x-4}{9}}\Leftrightarrow\dfrac{1}{2}\sqrt{x-1}-3=\dfrac{2}{3}\sqrt{x-1}\)
\(\Leftrightarrow\dfrac{1}{6}\sqrt{x-1}=-3\left(vôlí\right)\) vậy phương trình vô nghiệm
b) điều kiện xác định \(x\ge3\)
ta có : \(\sqrt{x^2-4x+4}+\sqrt{x^2+6x+9}=x-3\)
\(\Leftrightarrow\sqrt{\left(x-2\right)^2}+\sqrt{\left(x+3\right)^2}=x-3\) \(\Leftrightarrow\left|x-2\right|+\left|x+3\right|=x-3\)
\(\Leftrightarrow x-2+x+3=x-3\Leftrightarrow x=-4\left(L\right)\) vậy phương trình vô nghiệm
c) điều kiện xác định : \(\left[{}\begin{matrix}x\ge\dfrac{3}{2}\\x< 1\end{matrix}\right.\)
ta có : \(\sqrt{\dfrac{2x-3}{x-1}}=2\) \(\Leftrightarrow\dfrac{2x-3}{x-1}=4\Leftrightarrow2x-3=4x-4\)
\(\Leftrightarrow2x=1\Leftrightarrow x=\dfrac{1}{2}\left(tmđk\right)\) vậy \(x=\dfrac{1}{2}\)
a)\(\sqrt{x^2-2x+1}-\sqrt{x^2-4x+4}=x-3\)
\(\Leftrightarrow\left(\sqrt{x^2-2x+1}-3\right)-\left(\sqrt{x^2-4x+4}-2\right)=x-3-1\)
\(\Leftrightarrow\frac{x^2-2x+1-9}{\sqrt{x^2-2x+1}+3}-\frac{x^2-4x+4-4}{\sqrt{x^2-4x+4}+2}=x-4\)
\(\Leftrightarrow\frac{x^2-2x-8}{\sqrt{x^2-2x+1}+3}-\frac{x^2-4x}{\sqrt{x^2-4x+4}+2}-\left(x-4\right)=0\)
\(\Leftrightarrow\frac{\left(x+2\right)\left(x-4\right)}{\sqrt{x^2-2x+1}+3}-\frac{x\left(x-4\right)}{\sqrt{x^2-4x+4}+2}-\left(x-4\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(\frac{x+2}{\sqrt{x^2-2x+1}+3}-\frac{x}{\sqrt{x^2-4x+4}+2}-1\right)=0\)
Dễ thấy: \(\frac{x+2}{\sqrt{x^2-2x+1}+3}-\frac{x}{\sqrt{x^2-4x+4}+2}-1< 0\)
\(\Rightarrow x-1=0\Rightarrow x=1\)
b)\(\sqrt{x^2-6x+9}-\sqrt{x^2+6x+9}=1\)
\(\Leftrightarrow\left(\sqrt{x^2-6x+9}-\frac{7}{2}\right)-\left(\sqrt{x^2+6x+9}-\frac{5}{2}\right)=0\)
\(\Leftrightarrow\frac{x^2-6x+9-\frac{49}{4}}{\sqrt{x^2-6x+9}+\frac{7}{2}}-\frac{x^2+6x+9-\frac{25}{4}}{\sqrt{x^2+6x+9}+\frac{5}{2}}=0\)
\(\Leftrightarrow\frac{\frac{4x^2-24x-13}{4}}{\sqrt{x^2-6x+9}+\frac{7}{2}}-\frac{\frac{4x^2+24x+11}{4}}{\sqrt{x^2+6x+9}+\frac{5}{2}}=0\)
\(\Leftrightarrow\frac{\frac{\left(2x-13\right)\left(2x+1\right)}{4}}{\sqrt{x^2-6x+9}+\frac{7}{2}}-\frac{\frac{\left(2x+1\right)\left(2x+11\right)}{4}}{\sqrt{x^2+6x+9}+\frac{5}{2}}=0\)
\(\Leftrightarrow\left(2x+1\right)\left(\frac{\frac{2x-13}{4}}{\sqrt{x^2-6x+9}+\frac{7}{2}}-\frac{\frac{2x+11}{4}}{\sqrt{x^2+6x+9}+\frac{5}{2}}\right)=0\)
Dễ thấy: \(\frac{\frac{2x-13}{4}}{\sqrt{x^2-6x+9}+\frac{7}{2}}-\frac{\frac{2x+11}{4}}{\sqrt{x^2+6x+9}+\frac{5}{2}}< 0\)
\(\Rightarrow2x+1=0\Rightarrow x=-\frac{1}{2}\)
c)Áp dụng BĐT CAuchy-Schwarz ta có:
\(P^2=\left(\sqrt{x-2}+\sqrt{4-x}\right)^2\)
\(\le\left(1+1\right)\left(x-2+4-x\right)\)
\(=2\cdot\left(x-2+4-x\right)=2\cdot2=4\)
\(\Rightarrow P^2\le4\Rightarrow P\le2\)
Đăng 1 lúc mà nhiều thế. Lần sau đăng 1 câu thôi b.
b/ \(\sqrt{x^2-4x+5}+\sqrt{x^2-4x+8}+\sqrt{x^2-4x+9}=3+\sqrt{5}\)
\(\Leftrightarrow\sqrt{\left(x-2\right)^2+1}+\sqrt{\left(x-2\right)^2+4}+\sqrt{\left(x-2\right)^2+5}=3+\sqrt{5}\)
Ta có: \(VT\ge1+2+\sqrt{5}=3+\sqrt{5}\)
Dấu = xảy ra khi \(x=2\)
c/ \(\sqrt{2-x^2+2x}+\sqrt{-x^2-6x-8}=\sqrt{3-\left(x-1\right)^2}+\sqrt{1-\left(x+3\right)^2}\)
\(\le1+\sqrt{3}\)
Dấu = không xảy ra nên pt vô nghiệm
Câu d làm tương tự
\(a,\sqrt{x^2-4}-x^2+4=0\)
\(\Leftrightarrow\sqrt{x^2-4}=x^2-4\)
\(\Leftrightarrow x^2-4=\left(x-4\right)^2\)
\(\Leftrightarrow x^2-4-x^4+8x^2-16=0\)
\(\Leftrightarrow-x^4-7x^2-20=0\)
\(\Leftrightarrow-\left(x^4+7x^2+\frac{49}{4}\right)-\frac{31}{4}=0\)
\(\Leftrightarrow-\left(x^2+\frac{7}{2}\right)^2=\frac{31}{4}\)
\(\Leftrightarrow\left(x^2+\frac{7}{2}\right)=-\frac{31}{4}\)
\(\Rightarrow\)pt vô nghiệm
a/ ĐK: \(x \ge -1\). Đặt \(\sqrt{x+1}=a \ge 0\)
PT: \(\Leftrightarrow6a-3a-2a=5\)
\(\Leftrightarrow a=5\)
\(\Leftrightarrow x+1=15\Leftrightarrow x=24\) (nhận)
b,c: Hai ý này đều làm theo cách bình phương hoặc đưa về phương trình chứa dấu giá trị tuyệt đối được nhé.
b) Cách 1: ĐKXĐ: Tự tìm
\(\sqrt{x^{2}-4x+4}=2\Leftrightarrow x^{2}-4x+4=4\Leftrightarrow x(x-4)=0\)
\(\Leftrightarrow x=0\) hoặc \(x=4\) cả 2 cái này đều TMĐK
Cách 2: \((\sqrt{x^2-4x+4}=2)\)
\(\Leftrightarrow \sqrt{(x-2)^2}=2\)
\(\Leftrightarrow \mid x-2\mid=2\)
Với \(x\geq 2\) thì :
\(x-2=2 \Leftrightarrow x=4\) (nhận)
Với \(x<2\) thì
\(-x-2=2\Leftrightarrow x=0\) (nhận)
Vậy \(S={0;4}\)
c) Cách 1: \(\sqrt{x^{2}-6x+9}=x-2\Leftrightarrow \left\{\begin{matrix}x\geq 2 \\ x^{2}-6x+9=x^{2}-4x+4 \end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix}x\geq 2 \\ x=\frac{5}{2} \end{matrix}\right.\)
Nghiệm TMĐK
Cách 2: \((\sqrt{x^2-6x+9}=x-2)\)
\(\Leftrightarrow \mid x-3\mid =x-2\)
Với \(x\geq 3\) thì
\(x-3=x-2\Leftrightarrow 0x=-1\) ( vô lý)
Với \(x<3\) thì
\(-x+3=x-2\Leftrightarrow -2x=-5 \Leftrightarrow x=\frac{5}{2}\)
Vậy \(S={\frac{5}{2}}\)
d) ĐKXĐ: Tự tìm
\(\sqrt{x^{2}+4}=\sqrt{2x+3}\Leftrightarrow x^{2}+4=2x+3\Leftrightarrow x^{2}-2x+1=0\Leftrightarrow (x-1)^{2}=0\)
\(\Leftrightarrow x=1\)
e) ĐKXĐ: \(x\geq \frac{3}{2}\)
\(\frac{\sqrt{2x-3}}{\sqrt{x-1}}=2\Leftrightarrow \frac{2x-3}{x-1}=4\Rightarrow 2x-3=4x-4\Leftrightarrow x=\frac{1}{2}\)
Nghiệm không TMĐK.
Phương trình vô nghiệm.
f) ĐKXĐ: \(x\geq \frac{-15}{2}\)
\(x+\sqrt{2x+15}=0\Leftrightarrow 2x+2\sqrt{2x+15}=0\Leftrightarrow 2x+15+2\sqrt{2x+15}+1-16=0\)
\(\Leftrightarrow (\sqrt{2x+15}+1)^{2}-4^{2}=0\Leftrightarrow (\sqrt{2x+15}+5)(\sqrt{2x+15}-3)=0\)
\(\Leftrightarrow \sqrt{2x+15}-3=0\Leftrightarrow \sqrt{2x+15}=3\Leftrightarrow 2x+15=9\Leftrightarrow x=-3\) (TMĐK)
a) Đk: \(\left[{}\begin{matrix}x\le-1\\x\ge1\end{matrix}\right.\)
\(\sqrt{x^2-1}-x^2+1=0\)
\(\Leftrightarrow x^2-1-\sqrt{x^2-1}= 0\)
\(\Leftrightarrow\left(\sqrt{x^2-1}-1\right)\sqrt{x^2-1}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2-1}-1=0\\\sqrt{x^2-1}=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2-1}=1\\x^2-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2=2\left(1\right)\\x^2=1\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow x=\pm\sqrt{2}\left(N\right)\)
\(\left(2\right)\Leftrightarrow x=\pm1\left(N\right)\)
Kl: \(x=\pm\sqrt{2}\), \(x=\pm1\)
b) Đk: \(\left[{}\begin{matrix}x\le-2\\x\ge2\end{matrix}\right.\)
\(\sqrt{x^2-4}-x+2=0\)
\(\Leftrightarrow\sqrt{x^2-4}=x-2\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2-4=x^2-4x+4\\x\ge2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x=8\\x\ge2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=2\left(N\right)\\x\ge2\end{matrix}\right.\)
kl: x=2
c) \(\sqrt{x^4-8x^2+16}=2-x\)
\(\Leftrightarrow\sqrt{\left(x^2-4\right)^2}=2-x\)
\(\Leftrightarrow\left|x^2-4\right|=2-x\) (*)
Th1: \(x^2-4< 0\Leftrightarrow-2< x< 2\)
(*) \(\Leftrightarrow x^2-4=x-2\Leftrightarrow x^2-x-2=0\Leftrightarrow\left[{}\begin{matrix}x=2\left(L\right)\\x=-1\left(N\right)\end{matrix}\right.\)
Th2: \(x^2-4\ge0\Leftrightarrow\left[{}\begin{matrix}x\le-2\\x\ge2\end{matrix}\right.\)
(*)\(\Leftrightarrow x^2-4=2-x\Leftrightarrow x^2+x-6=0\Leftrightarrow\left[{}\begin{matrix}x=2\left(N\right)\\x=-3\left(N\right)\end{matrix}\right.\)
Kl: x=-3, x=-1,x=2
d) \(\sqrt{9x^2+6x+1}=\sqrt{11-6\sqrt{2}}\)
\(\Leftrightarrow\sqrt{\left(3x+1\right)^2}=\sqrt{\left(3-\sqrt{2}\right)^2}\)
\(\Leftrightarrow\left|3x+1\right|=3-\sqrt{2}\) (*)
Th1: \(3x+1\ge0\Leftrightarrow x\ge-\dfrac{1}{3}\)
(*) \(\Leftrightarrow3x+1=3-\sqrt{2}\Leftrightarrow x=\dfrac{2-\sqrt{2}}{3}\left(N\right)\)
Th2: \(3x+1< 0\Leftrightarrow x< -\dfrac{1}{3}\)
(*) \(\Leftrightarrow3x+1=-3+\sqrt{2}\Leftrightarrow x=\dfrac{-4+\sqrt{2}}{3}\left(N\right)\)
Kl: \(x=\dfrac{2-\sqrt{2}}{3}\), \(x=\dfrac{-4+\sqrt{2}}{3}\)
e) Đk: \(x\ge-\dfrac{3}{2}\)
\(\sqrt{4^2-9}=2\sqrt{2x+3}\) \(\Leftrightarrow\sqrt{7}=2\sqrt{2x+3}\) \(\Leftrightarrow7=8x+12\)
\(\Leftrightarrow8x=-5\Leftrightarrow x=-\dfrac{5}{8}\left(N\right)\)
kl: \(x=-\dfrac{5}{8}\)
f) Đk: x >/ 5
\(\sqrt{4x-20}+3\sqrt{\dfrac{x-5}{9}}-\dfrac{1}{3}\sqrt{9x-45}=4\)
\(\Leftrightarrow2\sqrt{x-5}+\sqrt{x-5}-\sqrt{x-5}=4\)
\(\Leftrightarrow2\sqrt{x-5}=4\)
\(\Leftrightarrow\sqrt{x-5}=2\)
\(\Leftrightarrow x-5=4\)
\(\Leftrightarrow x=9\left(N\right)\)
kl: x=9
Bài 1: Giải phương trình
a) ĐKXĐ: \(x\ge3\)
Ta có: \(\sqrt{100\cdot\left(x-3\right)}=\sqrt{20}\)
\(\Leftrightarrow\left|100\cdot\left(x-3\right)\right|=\left|20\right|\)
\(\Leftrightarrow100\cdot\left|x-3\right|=20\)
\(\Leftrightarrow\left|x-3\right|=\frac{1}{5}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=\frac{1}{5}\\x-3=-\frac{1}{5}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{16}{5}\left(nhận\right)\\x=\frac{14}{5}\left(loại\right)\end{matrix}\right.\)
Vậy: \(S=\left\{\frac{16}{5}\right\}\)
b) Ta có: \(\sqrt{\left(x-3\right)^2}=7\)
\(\Leftrightarrow\left|x-3\right|=7\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=7\\x-3=-7\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=10\\x=-4\end{matrix}\right.\)
Vậy: S={10;-4}
c) Ta có: \(\sqrt{4x^2+4x+1}=6\)
\(\Leftrightarrow\sqrt{\left(2x+1\right)^2}=6\)
\(\Leftrightarrow\left|2x+1\right|=6\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+1=6\\2x+1=-6\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}2x=5\\2x=-7\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{5}{2}\\x=\frac{-7}{2}\end{matrix}\right.\)
Vậy: \(S=\left\{\frac{5}{2};\frac{-7}{2}\right\}\)
a) `sqrt(x^2-6x _9) = 4-x`
`<=> sqrt[(x-3)^2] =4-x`
`<=> |x-3| =4-x ( đk :x<=4)`
`<=> |x-3| = |4-x|`
`<=> [(x-3 =4-x),(x-3 = x-4):}`
`<=>[(x = 7/2(t//m)),(0=-1(vl)):}`
Vậy `S = {7/2}`
b) `sqrt(x^2 -9) + sqrt(x^2 -6x +9) =0(đk : x>=3(hoặc) x<=-3)`
`<=>sqrt(x^2 -9) =- sqrt(x^2 -6x +9) `
`<=>(sqrt(x^2 -9))^2 =(- sqrt(x^2 -6x +9))^2`
`<=> x^2 -9 = x^2 -6x +9`
`<=> 6x = 9+9 =18`
`<=> x=3(t//m)`
Vậy `S={3}`
c) `sqrt(x^2 -2x+1) + sqrt(x^2-4x+4) =3`
`<=> sqrt[(x-1)^2] +sqrt[(x-2)^2] =3`
`<=> |x-1| +|x-2| =3`
xét `x<1 =>{(|x-1| =1-x ),(|x-2|=2-x):}`
`=> 1-x +2-x =3`
`=> x = 0(t//m)`
xét `1<=x<2 => {(|x-1|=x-1),(|x-2|= 2-x):}`
`=> x-1 +2-x =3`
`=>1=3 (vl)`
xét `x>=2 => {(|x-1| =x-1),(|x-2|=x-2):}`
`=> x-1+x-2 =3`
`=> x=3(t//m)`
Vậy `S = {0;3}`