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\(\Leftrightarrow\sqrt[3]{3x+1}+\sqrt[3]{5-x}=\sqrt[3]{4x-3}+\sqrt[3]{9-2x}\)
Đặt \(\left\{{}\begin{matrix}\sqrt[3]{3x+1}=a\\\sqrt[3]{5-x}=b\\\sqrt[3]{4x-3}=c\\\sqrt[3]{9-2x}=d\end{matrix}\right.\)
Ta được: \(\left\{{}\begin{matrix}a+b=c+d\\a^3+b^3=c^3+d^3\end{matrix}\right.\)
TH1:
Nếu \(a+b=c+d=0\Leftrightarrow\sqrt[3]{3x+1}+\sqrt[3]{5-x}=\sqrt[3]{4x-3}+\sqrt[3]{9-2x}=0\)
\(\Rightarrow\left\{{}\begin{matrix}3x+1=-\left(5-x\right)\\4x-3=-\left(9-2x\right)\end{matrix}\right.\) \(\Rightarrow x=-3\)
TH2: nếu \(a+b=c+d\ne0\)
\(a+b=c+d\Leftrightarrow\left(a+b\right)^3=\left(c+d\right)^3\)
\(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)=c^3+d^3+3cd\left(c+d\right)\)
\(\Leftrightarrow ab\left(a+b\right)=cd\left(c+d\right)\) (do \(a^3+b^3=c^3+d^3\))
\(\Leftrightarrow ab=cd\) (do \(a+b=c+d\ne0\))
\(\Leftrightarrow\sqrt[3]{\left(3x+1\right)\left(5-x\right)}=\sqrt[3]{\left(4x-3\right)\left(9-2x\right)}\)
\(\Leftrightarrow\left(3x+1\right)\left(5-x\right)=\left(4x-3\right)\left(9-2x\right)\)
\(\Leftrightarrow5x^2-28x+32=0\)
\(\Rightarrow\left[{}\begin{matrix}x=4\\x=\dfrac{8}{5}\end{matrix}\right.\)
Vậy \(x=\left\{-3;4;\dfrac{8}{5}\right\}\)
Cái cuối này căn bậc 2 hay căn bậc 3 em? Căn bậc 2 thì hơi nghi ngờ về khả năng giải được của pt này.
a)√x2−9 - 3√x−3 =0
<=> (√x-3)(√x+3)-3√x-3=0
<=> (√x-3)(√x+3-3)=0
<=> (√x-3)√x=0
<=> √x-3=0
<=>x=9
b)√4x2−12x+9=x - 3
<=> √(2x -3)2 =x-3
<=> 2x-3=x-3
<=>2x-x=-3+3
<=>x=0
c)√x2+6x+9=3x-1
<=> √(x+3)2 =3x-1
<=> x+3=3x-1
<=> -2x=-4
<=> x=2
Nhớ cho mình 1 tim nha bạn
Sau em nên gõ các kí hiệu toán học ở phần Σ để mọi người dễ dàng đọc hơn nhé.
1.
\(\sqrt{50}-3\sqrt{8}+\sqrt{32}=5\sqrt{2}-6\sqrt{2}+4\sqrt{2}=3\sqrt{2}\)
2.
a, ĐK: \(x\in R\)
\(pt\Leftrightarrow\sqrt{\left(x-2\right)^2}=1\)
\(\Leftrightarrow\left|x-2\right|=1\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2=1\\x-2=-1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\\x=1\end{matrix}\right.\)
b, ĐK: \(x\ge3\)
\(pt\Leftrightarrow\sqrt{x-3}\left(\sqrt{x}-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-3}=0\\\sqrt{x}-1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\left(tm\right)\\x=1\left(l\right)\end{matrix}\right.\)
ĐKXĐ: \(x\ge1\)
\(\sqrt{5x-1}=\sqrt{3x-2}+\sqrt{x-1}\)
\(\Leftrightarrow5x-1=3x-2+x-1+2\sqrt{\left(3x-2\right)\left(x-1\right)}\)
\(\Leftrightarrow x+2=2\sqrt{\left(3x-2\right)\left(x-1\right)}\)
\(\Leftrightarrow x^2+4x+4=4\left(3x-2\right)\left(x-1\right)\)
\(\Leftrightarrow11x^2-24x+4=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{2}{11}\left(loại\right)\\x=2\end{matrix}\right.\)
1. 3x( x - 2 ) - ( x - 2 ) = 0
<=> ( x-2).(3x-1) = 0 => x = 2 hoặc x = \(\dfrac{1}{3}\)
2. x( x-1 ) ( x2 + x + 1 ) - 4( x - 1 )
<=> ( x - 1 ).( x (x^2 + x + 1 ) - 4 ) = 0
(phần này tui giải được x = 1 thôi còn bên kia giải ko ra nha )
3 \(\left\{{}\begin{matrix}\sqrt{5}x-2y=7\\\sqrt{5}x-5y=10\end{matrix}\right.\)<=> \(\left\{{}\begin{matrix}y=-1\\x=\sqrt{5}\end{matrix}\right.\)
\(1. 3x^2 - 7x +2=0\)
=>\(Δ=(-7)^2 - 4.3.2\)
\(= 49-24 = 25\)
Vì 25>0 suy ra phương trình có 2 nghiệm phân biệt:
\(x_1\)=\(\dfrac{-\left(-7\right)+\sqrt{25}}{2.3}=\dfrac{7+5}{6}=2\)
\(x_2\)=\(\dfrac{-\left(-7\right)-\sqrt{25}}{2.3}=\dfrac{7-5}{6}=\dfrac{1}{3}\)
ĐKXĐ: \(-\dfrac{1}{3}\le x\le6\)
\(\left(\sqrt{3x+1}-4\right)+\left(1-\sqrt{6-x}\right)+\left(3x^2-14x-5\right)=0\)
\(\Leftrightarrow\dfrac{3\left(x-5\right)}{\sqrt{3x+1}+4}+\dfrac{x-5}{1+\sqrt{6-x}}+\left(x-5\right)\left(3x+1\right)=0\)
\(\Leftrightarrow\left(x-5\right)\left(\dfrac{3}{\sqrt{3x+1}+4}+\dfrac{1}{1+\sqrt{6-x}}+3x+1\right)=0\)
\(\Leftrightarrow x-5=0\) (do \(\dfrac{3}{\sqrt{3x+1}+4}+\dfrac{1}{1+\sqrt{6-x}}+3x+1>0;\forall x\))
\(\Rightarrow x=5\)
ĐKXĐ: \(\left\{{}\begin{matrix}3x+1>=0\\6-x>=0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x>=-\dfrac{1}{3}\\x< =6\end{matrix}\right.\)
\(\sqrt{3x+1}-\sqrt{6-x}+3x^2-14x-8=0\)
=>\(\sqrt{3x+1}-4+1-\sqrt{6-x}+3x^2-14x-5=0\)
=>\(\dfrac{3x+1-16}{\sqrt{3x+1}+4}+\dfrac{1-6+x}{1+\sqrt{6-x}}+3x^2-15x+x-5=0\)
=>\(\dfrac{3\cdot\left(x-5\right)}{\sqrt{3x+1}+4}+\dfrac{x-5}{\sqrt{6-x}+1}+\left(x-5\right)\left(3x+1\right)=0\)
=>\(\left(x-5\right)\left(\dfrac{3}{\sqrt{3x+1}+4}+\dfrac{1}{\sqrt{6-x}+1}+3x+1\right)=0\)
=>x-5=0
=>x=5(nhận)
a) \(3x-2\sqrt{x-1}=4\) (ĐK: x ≥ 1)
\(\Rightarrow3x-2\sqrt{x-1}-4=0\)
\(\Rightarrow3x-6-2\sqrt{x-1}+2=0\)
\(\Rightarrow3\left(x-2\right)-2\left(\sqrt{x-1}-1\right)=0\)
\(\Rightarrow3\left(x-2\right)-2.\dfrac{x-2}{\sqrt{x-1}+1}=0\)
\(\Rightarrow\left(x-2\right)\left[3-\dfrac{2}{\sqrt{x-1}+1}\right]=0\)
*TH1: x = 2 (t/m)
*TH2: \(3-\dfrac{2}{\sqrt{x-1}+1}=0\)
\(\Rightarrow3=\dfrac{2}{\sqrt{x-1}+1}\)
\(\Rightarrow3\sqrt{x-1}+3=2\)
\(\Rightarrow3\sqrt{x-1}=-1\) (vô lí)
Vậy S = {2}
b) \(\sqrt{4x+1}-\sqrt{x+2}=\sqrt{3-x}\) (ĐK: \(-\dfrac{1}{4}\le x\le3\) )
\(\Rightarrow\sqrt{4x+1}-3-\sqrt{x+2}+2-\sqrt{3-x}+1=0\)
\(\Rightarrow\dfrac{4x-8}{\sqrt{4x+1}+3}-\dfrac{x-2}{\sqrt{x+2}+2}+\dfrac{x-2}{\sqrt{3-x}+1}=0\)
\(\Rightarrow\left(x-2\right)\left(\dfrac{4}{\sqrt{4x+1}+3}-\dfrac{1}{\sqrt{x+2}+2}+\dfrac{1}{\sqrt{3-x}+1}\right)=0\)
=> x = 2
\(a,3x-2\sqrt{x-1}=4\left(x\ge1\right)\\ \Leftrightarrow-2\sqrt{x-1}=4-3x\\ \Leftrightarrow4\left(x-1\right)=16-24x+9x^2\\ \Leftrightarrow9x^2-28x+20=0\\ \Leftrightarrow\left(x-2\right)\left(9x-10\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=2\left(tm\right)\\x=\dfrac{10}{9}\left(tm\right)\end{matrix}\right.\)
\(b,\sqrt{4x+1}-\sqrt{x+2}=\sqrt{3-x}\left(-\dfrac{1}{4}\le x\le3\right)\\ \Leftrightarrow4x+1+x+2-2\sqrt{\left(4x+1\right)\left(x+2\right)}=3-x\\ \Leftrightarrow-2\sqrt{\left(4x+1\right)\left(x+2\right)}=2-6x\\ \Leftrightarrow\sqrt{4x^2+9x+2}=3x-1\\ \Leftrightarrow4x^2+9x+2=9x^2-6x+1\\ \Leftrightarrow5x^2-15x-1=0\\ \Leftrightarrow\Delta=225+20=245\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{15-\sqrt{245}}{10}=\dfrac{15-7\sqrt{5}}{10}\left(ktm\right)\\x=\dfrac{15+\sqrt{245}}{10}=\dfrac{15+7\sqrt{5}}{10}\left(tm\right)\end{matrix}\right.\Leftrightarrow x=\dfrac{15+7\sqrt{5}}{10}\)
ĐKXĐ: \(-\dfrac{1}{3}\le x\le4\)
\(\Leftrightarrow x+5=\sqrt{3x+1}+2\sqrt{4-x}\)
Ta có:
\(VP=1.\sqrt{3x+1}+2.\sqrt{4-x}\le\dfrac{1}{2}\left(1+3x+1\right)+\dfrac{1}{2}\left(4+4-x\right)=x+5\)
\(\Rightarrow VP\le VT\)
Dấu "=" xảy ra khi và chỉ khi: \(\left\{{}\begin{matrix}\sqrt{3x+1}=1\\\sqrt{4-x}=2\end{matrix}\right.\) \(\Leftrightarrow x=0\)