a,ĐKXĐ:\(x\ge2\)

\(4\sqrt{x-2}+\sqrt{9x-18}-\sqrt{\dfrac{x-2}{4}}=26\\ \Leftrightarrow4\sqrt{x-2}+3\sqrt{x-2}-\dfrac{\sqrt{x-2}}{2}=26\\ \Leftrightarrow8\sqrt{x-2}+6\sqrt{x-2}-\sqrt{x-2}=52\\ \Leftrightarrow13\sqrt{x-2}=52\\ \Leftrightarrow\sqrt{x-2}=4\\ \Leftrightarrow x-2=16\\ \Leftrightarrow x=18\left(tm\right)\)

b,ĐKXĐ:\(x\in R\)

\(3x+\sqrt{4x^2-8x+4}=1\\ \Leftrightarrow2\sqrt{x^2-2x+1}=1-3x\\ \Leftrightarrow\left|x-1\right|=\dfrac{1-3x}{2}\\ \Leftrightarrow\left[{}\begin{matrix}x-1=\dfrac{1-3x}{2}\\x-1=\dfrac{3x-1}{2}\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}2x-2=1-3x\\2x-2=3x-1\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{3}{5}\left(tm\right)\\x=-1\left(tm\right)\end{matrix}\right.\)

c, ĐKXĐ:\(x\ge0\)

\(\left(2\sqrt{x}+1\right)\left(\sqrt{x}-2\right)=7\\ \Leftrightarrow\sqrt{x}\left(2\sqrt{x}+1\right)-2\left(2\sqrt{x}+1\right)=7\\ \Leftrightarrow2x+\sqrt{x}-4\sqrt{x}-2=7\\ \Leftrightarrow2x-3\sqrt{x}-9=0\\ \Leftrightarrow\left(2x+3\sqrt{x}\right)-\left(6\sqrt{x}+9\right)=0\\ \Leftrightarrow\sqrt{x}\left(2\sqrt{x}+3\right)-3\left(2\sqrt{x}+3\right)=0\\ \Leftrightarrow\left(\sqrt{x}-3\right)\left(2\sqrt{x}+3\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}\sqrt{x}=3\\2\sqrt{x}=-3\left(vô.lí\right)\end{matrix}\right.\\ \Leftrightarrow x=9\left(tm\right)\)

Lời giải:

ĐKXĐ: $x\geq \frac{1}{2}$

PT $\Leftrightarrow (3x^2-10x-25)=2(x+3)(\sqrt{2x-1}-3)$

$\Leftrightarrow (x-5)(3x+5)=2(x+3).\frac{2(x-5)}{\sqrt{2x-1}+3}$

\(\Leftrightarrow (x-5)\left[(3x+5)-\frac{4(x+3)}{\sqrt{2x-1}+3}\right]=0\)

Xét biểu thức trong ngoặc vuông:

\(\Leftrightarrow (3x+5)(\sqrt{2x-1}+3)=4(x+3)\)

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

Dễ thấy điều này vô lý vì với $x\geq \frac{1}{2}$ thì vế trái không âm còn vế phải âm.

Vậy $x-5=0\Leftrightarrow x=5$

a.

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

Đặt \(\sqrt[3]{3\left(x+1\right)+2}=y\) ta được:

\(\left\{{}\begin{matrix}3y=\left(x+1\right)^3-2\\3\left(x+1\right)+2=y^3\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}3y+2=\left(x+1\right)^3\\3\left(x+1\right)+2=y^3\end{matrix}\right.\)

\(\Rightarrow\left(x+1\right)^3-y^3=3y-3\left(x+1\right)\)

\(\Leftrightarrow\left(x+1-y\right)\left[\left(x+1\right)^2+y\left(x+1\right)+y^2+3\right]=0\)

\(\Leftrightarrow x+1=y\)

\(\Leftrightarrow\left(x+1\right)^3=y^3\)

\(\Leftrightarrow\left(x+1\right)^3=3\left(x+1\right)+2\)

\(\Leftrightarrow x^3+3x^2-4=0\)

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

b.

\(\Leftrightarrow8x^3-\left(6x+1\right)+2x-\sqrt[3]{6x+1}=0\)

Đặt \(\left\{{}\begin{matrix}2x=a\\\sqrt[3]{6x+1}=b\end{matrix}\right.\) ta được:

\(a^3-b^3+a-b=0\)

\(\Leftrightarrow\left(a-b\right)\left(a^2+ab+b^2+1\right)=0\)

\(\Leftrightarrow a=b\)

\(\Leftrightarrow2x=\sqrt[3]{6x+1}\)

\(\Leftrightarrow8x^3-6x-1=0\)

Đặt \(f\left(x\right)=8x^3-6x-1\)

\(f\left(x\right)\) là hàm đa thức nên liên tục trên R, đồng thời \(f\left(x\right)\) bậc 3 nên có tối đa 3 nghiệm

\(f\left(-1\right)=-3< 0\) ; \(f\left(-\dfrac{1}{2}\right)=1>0\) \(\Rightarrow f\left(-1\right).f\left(-\dfrac{1}{2}\right)< 0\)

\(\Rightarrow f\left(x\right)\) có 1 nghiệm thuộc \(\left(-1;-\dfrac{1}{2}\right)\) (1)

\(f\left(0\right)=-1\Rightarrow f\left(0\right).f\left(-\dfrac{1}{2}\right)< 0\Rightarrow f\left(x\right)\) có 1 nghiệm thuộc \(\left(-\dfrac{1}{2};0\right)\) (2)

\(f\left(1\right)=1\Rightarrow f\left(0\right).f\left(1\right)< 0\Rightarrow f\left(x\right)\) có 1 nghiệm thuộc \(\left(0;1\right)\) (3)

Từ (1);(2);(3) \(\Rightarrow\) cả 3 nghiệm của \(f\left(x\right)\) đều thuộc \(\left(-1;1\right)\)

Do đó, ta chỉ cần tìm nghiệm của \(f\left(x\right)\) với \(x\in\left(-1;1\right)\)

Do \(x\in\left(-1;1\right)\), đặt \(x=cosu\)

\(\Rightarrow8cos^3u-6cosu-1=0\)

\(\Leftrightarrow2\left(4cos^3u-3cosu\right)=1\)

\(\Leftrightarrow2cos3u=1\)

\(\Leftrightarrow cos3u=\dfrac{1}{2}\)

\(\Rightarrow\left[{}\begin{matrix}3u=\dfrac{\pi}{3}+k2\pi\\3u=-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}u=\dfrac{\pi}{9}+\dfrac{k2\pi}{3}\\u=-\dfrac{\pi}{9}+\dfrac{k2\pi}{3}\end{matrix}\right.\)

Vậy nghiệm của pt là: \(x=cosu=\left\{cos\left(\dfrac{\pi}{9}\right);cos\left(\dfrac{5\pi}{9}\right);cos\left(\dfrac{7\pi}{9}\right)\right\}\)

Liên hợp đi

\(1.\sqrt{16-8x+x^2}=4-x\)

\(\sqrt{\left(4-x\right)^2}=4-x\)

\(4-x-4+x=0\)

= 0 phương trình vô nghiệm.

\(2.\sqrt{4x^2-12x+9}=2x-3\)

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

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

= 0 phương trình vô nghiệm.

a: Ta có: \(\sqrt{16-8x+x^2}=4-x\)

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

hay \(x\le4\)

b: Ta có: \(\sqrt{4x^2-12x+9}=2x-3\)

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

hay \(x\ge\dfrac{3}{2}\)

a/ \(\sqrt{16-8x+x^2}=4-x\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le4\\\sqrt{\left(4-x\right)^2}=4-x\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le4\\\left|4-x\right|=4-x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x\le4\\\left[{}\begin{matrix}4-x=4-x\left(loại\right)\\4-x=x-4\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow x=4\)

Vậy...

b/ \(\sqrt{4x^2-12x+9}=2x-3\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{2}{3}\\\sqrt{\left(2x-3\right)^2}=2x-3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{2}{3}\\\left[{}\begin{matrix}2x-3=2x-3\left(loại\right)\\2x-3=3-2x\end{matrix}\right.\end{matrix}\right.\)

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

Vậy...

cho mik hỏi rằng là 3x2 + 4x = 0 hay  3x² + 4x = 0

ông ơi mấy bài này bấm máy tính là ra mà ông

a) \(3x^2+4x=0\Leftrightarrow\left(3x+4\right)x=0\Leftrightarrow\left[{}\begin{matrix}x=0\\3x+4=0\Leftrightarrow x=-\dfrac{4}{3}\end{matrix}\right.\)

   ➤\(x\in\left\{0;-\dfrac{4}{3}\right\}\)

b) \(-2x^2-8=0\Leftrightarrow-2x^2+\left(-2\right)\cdot4=0\)

                           \(\Leftrightarrow\left(x^2+4\right)\cdot\left(-2\right)=0\\ \Leftrightarrow x^2+4=0\\\Rightarrow x^2=\varnothing\Leftrightarrow x=\varnothing \) 

                          vì với mọi x, ta luôn đúng với: \(x^2\ge0\Leftrightarrow x^2+4\ge4>0\)

➤\(x=\varnothing\)

c)\(2x^2-7x^2+5=0\)

+) \(a+b+c=2+\left(-7\right)+5=7-7=0\)

Do đó, phương trình có 2 nghiệm sau:

\(x=1\) và \(x=\dfrac{5}{2}=2,5\)

➤\(x\in\left\{1;2,5\right\}\)

d) \(x^2-8x-48=0\)

+)\(\Delta=\left(-8\right)^2-4\cdot1\cdot\left(-48\right)=64+192=266>0\)

\(\Leftrightarrow\sqrt{\Delta}=\sqrt{266}\)

➢Do đó, ta có: \(\left[{}\begin{matrix}x=\dfrac{\sqrt{266}-\left(-8\right)}{2\cdot2}=\dfrac{\sqrt{266}+8}{4}\\x=\dfrac{-\sqrt{266}-\left(-8\right)}{2\cdot2}=\dfrac{8-\sqrt{266}}{4}\end{matrix}\right.\)

➤ \(x\in\left\{\dfrac{8+\sqrt{266}}{4};\dfrac{8-\sqrt{266}}{4}\right\}\)