a) Để biểu thức xác định thì \(3x^2+2\ne0\forall x\in R\)

vậy với mọi x thì biểu thức trên luôn xác định.

b) Để .......

\(\left\{{}\begin{matrix}2x+5\ge0\\x-1>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ge-\frac{5}{2}\\x>1\end{matrix}\right.\)

vậy biểu thức trên xác định khi x>1.

c) Để ..........

\(\left\{{}\begin{matrix}x+1\ge0\\x^2-2x\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\\left\{{}\begin{matrix}x\ne0\\x\ne2\end{matrix}\right.\end{matrix}\right.\)

Vậy để biểu thức xđ khi \(x\in[-1;+\infty)\backslash\left\{0;2\right\}\)

d) Để ........

\(\left\{{}\begin{matrix}2x+3\ge0\\5-x\ge\\2-\sqrt{5-x}\ne0\end{matrix}\right.0\Leftrightarrow\left\{{}\begin{matrix}x\ge-\frac{3}{2}\\x\le5\\x\ne1\end{matrix}\right.\)

Vậy để btxđ khi \(x\in\left[-\frac{3}{2};5\right]\backslash\left\{1\right\}\)

e) Để ......

\(\left\{{}\begin{matrix}x+2\ge0\\3-2x\ge0\\\left|x\right|-1\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ge-2\\x\le\\\left\{{}\begin{matrix}x\ne1\\x\ne-1\end{matrix}\right.\end{matrix}\right.\frac{3}{2}\)

Vậy để btxđ khi ....

Rồi yêu cầu đề bài đâu bạn. ?

\(x\in\left\{\varnothing\right\}\)

Tổng hợp hệ pt

Bài 2:
ĐKXĐ: $6\geq x\geq \frac{-1}{3}$
PT $\Leftrightarrow (\sqrt{3x+1}-4)+(1-\sqrt{6-x})+(3x^2-14x-5)=0$

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

Với $x$ thuộc đkxđ, dễ thấy biểu thức trong ngoặc vuông $>0$

$\Rightarrow x-5=0$

$\Leftrightarrow x=5$

Bài 3:

PT $3x=\sqrt{x^2+12}-\sqrt{x^2+5}+5>0$

$\Rightarrow x>0$

Lại có:

PT $\Leftrightarrow \sqrt{x^2+12}-4=3(x-2)+(\sqrt{x^2+5}-3)$

$\Leftrightarrow \frac{x^2-4}{\sqrt{x^2+12}+4}=3(x-2)+\frac{x^2-4}{\sqrt{x^2+5}+3}$

$\Leftrightarrow (x-2)\left[\frac{x+2}{\sqrt{x^2+12}+4}-3-\frac{x+2}{\sqrt{x^2+5}+3}\right]=0$

Với $x>0$, dễ thấy:
$\frac{x+2}{\sqrt{x^2+5}+3}+3>\frac{x+2}{\sqrt{x^2+12}+4}$ nên biểu thức trong ngoặc vuông âm.

Do đó $x-2=0\Leftrightarrow x=2$ (tm)

a/ \(\left(2x-3\right)\left(3x-4\right)\left(5x+2\right)>0\)

\(\Rightarrow\left[{}\begin{matrix}-\frac{2}{3}< x< \frac{4}{3}\\x>\frac{3}{2}\end{matrix}\right.\)

b/ \(\Leftrightarrow24x^2-10x-25< 0\)

\(\Rightarrow-\frac{5}{6}< x< \frac{5}{4}\)

c/ \(\frac{4x\left(3x+2\right)}{2x+5}>0\Rightarrow\left[{}\begin{matrix}-\frac{5}{2}< x< -\frac{2}{3}\\x>0\end{matrix}\right.\)

d/ \(\Leftrightarrow\frac{3x+2}{2x-5}-\frac{2x-5}{3x+2}\ge0\)

\(\Leftrightarrow\frac{\left(3x+2\right)^2-\left(2x-5\right)^2}{\left(2x-5\right)\left(3x+2\right)}\ge0\)

\(\Leftrightarrow\frac{\left(5x-2\right)\left(x+7\right)}{\left(2x-5\right)\left(3x+2\right)}\ge0\Rightarrow\left[{}\begin{matrix}x\le-7\\-\frac{2}{3}< x\le\frac{2}{5}\\x>\frac{5}{2}\end{matrix}\right.\)

Do \(1\le x\le2\Rightarrow\left(x-1\right)\left(x-2\right)\le0\)

\(\Leftrightarrow x^2+2\le3x\)

Tương tự \(y^2+2\le3y\)

Do đó:

\(P=\frac{x+2y}{x^2+2+3y+3}+\frac{2x+y}{y^2+2+3x+3}+\frac{1}{4\left(x+y-1\right)}\ge\frac{x+2y}{3x+3y+3}+\frac{2x+y}{3x+3y+3}+\frac{1}{4\left(x+y-1\right)}\)

\(P\ge\frac{3x+3y}{3x+3y+3}+\frac{1}{4\left(x+y-1\right)}=\frac{x+y}{x+y+1}+\frac{1}{4\left(x+y-1\right)}\)

Đặt \(x+y=t\Rightarrow2\le t\le4\)

\(\Rightarrow P\ge\frac{t}{t+1}+\frac{1}{4t-4}=\frac{t}{t+1}+\frac{1}{4t-4}-\frac{7}{8}+\frac{7}{8}\)

\(P\ge\frac{\left(t-3\right)^2}{8\left(t^2-1\right)}+\frac{7}{8}\ge\frac{7}{8}\)

\(P_{min}=\frac{7}{8}\) khi \(t=3\) hay \(\left(x;y\right)=\left(1;2\right);\left(2;1\right)\)

Nguyễn Việt Lâm a giúp e vs a

đéo biết

a/ \(\Leftrightarrow x^2+5x-2-2\sqrt[3]{x^2+5x-2}+4=0\)

Đặt \(\sqrt[3]{x^2+5x-2}=a\)

\(a^3-2a+4=0\)

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

\(\Rightarrow\sqrt[3]{x^2+5x-2}=-2\Rightarrow x^2+5x+6=0\Rightarrow...\)

b/ ĐKXĐ:...

\(\Leftrightarrow-3\left(-x^2+4x+10\right)-5\sqrt{-x^2+4x+10}+42=0\)

Đặt \(\sqrt{-x^2+4x+10}=a\ge0\)

\(-3a^2-5a+42=0\Rightarrow\left[{}\begin{matrix}a=3\\a=-\frac{14}{3}\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt{x^2+4x+10}=3\Rightarrow x^2-4x-1=0\Rightarrow...\)

c/ ĐKXĐ: ...

\(\Leftrightarrow x^2+3x+3\sqrt{x^2+3x}-10=0\)

Đặt \(\sqrt{x^2+3x}=a\ge0\)

\(a^2+3a-10=0\Rightarrow\left[{}\begin{matrix}a=2\\a=-5\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt{x^2+3x}=2\Rightarrow x^2+3x-4=0\)

d/ ĐKXĐ: \(-1\le x\le2\)

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

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

\(\Leftrightarrow2+x-x^2+\sqrt{2+x-x^2}-2=0\)

Đặt \(\sqrt{2+x-x^2}=a\ge0\)

\(a^2+a-2=0\Rightarrow\left[{}\begin{matrix}a=1\\a=-2\left(l\right)\end{matrix}\right.\)

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

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

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

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

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

câu 1 đề sai hay vô nghiệm ko bt

câu 2: pt thứ 2 thiếu

nếu chưa ai làm chiều học về mk sẽ làm