a: \(\text{Δ}=\left(2m+1\right)^2-4m\left(m+3\right)\)

\(=4m^2+4m+1-4m^2-12m\)

\(=-8m+1\)

Để phương trình có hai nghiệm phân biệt thì Δ>0

\(\Leftrightarrow-8m+1>0\)

\(\Leftrightarrow-8m>-1\)

hay \(m< \dfrac{1}{8}\)

Đề sai rồi bạn

a:Sửa đề: \(\dfrac{3}{5x-1}+\dfrac{2}{3-x}=\dfrac{4}{\left(1-5x\right)\left(x-3\right)}\)

=>3x-9-10x+2=-4

=>-7x-7=-4

=>-7x=3

=>x=-3/7

b: =>\(\dfrac{5-x}{4x\left(x-2\right)}+\dfrac{7}{8x}=\dfrac{x-1}{2x\left(x-2\right)}+\dfrac{1}{8\left(x-2\right)}\)

=>\(2\left(5-x\right)+7\left(x-2\right)=4\left(x-1\right)+x\)

=>10-2x+7x-14=4x-4+x

=>5x-4=5x-4

=>0x=0(luôn đúng)

Vậy: S=R\{0;2}

a: Khi m = -4 thì:

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

\(\Leftrightarrow x^2-5x-6=0\)

\(\Delta=\left(-5\right)^2-5\cdot1\cdot\left(-6\right)=49\Rightarrow\sqrt{\Delta}=\sqrt{49}=7>0\)

Pt có 2 nghiệm phân biệt:

\(x_1=\dfrac{5+7}{2}=6;x_2=\dfrac{5-7}{2}=-1\)

b: \(\Delta=\left(-5\right)^2-4\left(m-2\right)=25-4m+8=33-4m\)

Theo viet:

\(x_1+x_2=-\dfrac{b}{a}=5\)

\(x_1x_2=\dfrac{c}{a}=m-2\)

Để pt có 2 nghiệm dương phân biệt:

\(\Leftrightarrow\left\{{}\begin{matrix}\Delta>0\\x_1+x_2>0\\x_1x_2>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}33-4m>0\\5>0\left(TM\right)\\m-2>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m< \dfrac{33}{4}\\x>2\end{matrix}\right.\Leftrightarrow m=2< m< \dfrac{33}{4}\)

Vậy \(2< m< \dfrac{33}{4}\) thì pt có 2 nghiệm dương phân biệt.

Theo đầu bài: \(\dfrac{1}{\sqrt{x_1}}+\dfrac{1}{\sqrt{x_2}}=\dfrac{3}{2}\)

\(\Leftrightarrow\sqrt{x_1}+\sqrt{x_2}=\dfrac{3}{2}\left(\sqrt{x_1x_2}\right)\)

\(\Leftrightarrow\left(\sqrt{x_1}+\sqrt{x_2}\right)^2=\dfrac{9}{4}x_1x_2\)

\(\Leftrightarrow x_1+2\sqrt{x_1x_2}+x_2=\dfrac{9}{4}x_1x_2\)

\(\Leftrightarrow x_1+x_2+2\sqrt{x_1x_2}=\dfrac{9}{4}x_1x_2\)

\(\Leftrightarrow5+2\sqrt{x_1x_2}=\dfrac{9}{4}\left(m-2\right)\)

\(\Leftrightarrow\dfrac{9}{4}\left(m-2\right)-2\sqrt{m-2}-5=0\)

Đặt \(\sqrt{m-2}=t\Rightarrow m-2=t^2\)

\(\Rightarrow\dfrac{9}{4}t^2-2t-5=0\)

\(\Leftrightarrow\dfrac{9}{4}t^2-2+\left(-5\right)=0\)

\(\Leftrightarrow\left(t-2\right)\left(9t+10\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t-2=0\\9t+10=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}t=2\left(TM\right)\\t=-\dfrac{10}{9}\left(\text{loại}\right)\end{matrix}\right.\)

Trả ẩn:

\(\sqrt{m-2}=2\)

\(\Rightarrow m-2=4\)

\(\Rightarrow m=6\)

Vậy m = 6 thì x₁ , x₂ thoả mãn hệ thức \(2\left(\dfrac{1}{\sqrt{x_1}}+\dfrac{1}{\sqrt{x_2}}\right)=\dfrac{3}{2}\).

a:=>3x=15

=>x=5

b: =>8-11x<52

=>-11x<44

=>x>-4

c: \(VT=\left(\dfrac{x^2-\left(x-6\right)^2}{x\left(x+6\right)\left(x-6\right)}\right)\cdot\dfrac{x\left(x+6\right)}{2x-6}+\dfrac{x}{6-x}\)

\(=\dfrac{12x-36}{2x-6}\cdot\dfrac{1}{x-6}-\dfrac{x}{x-6}=\dfrac{6}{x-6}-\dfrac{x}{x-6}=-1\)

\(\Leftrightarrow\sin x+\dfrac{\pi}{3}=\dfrac{\pi}{2}+k2\pi\)

\(\Leftrightarrow2x=\dfrac{\pi}{6}+k2\pi\)

\(\Leftrightarrow x=\dfrac{\pi}{12}+k\pi\left(k\in Z\right)\)

Vì x ∈ \(\left[-\pi;-2\pi\right]\) ta có:

\(-2\pi\le\dfrac{\pi}{12}+k\pi\le-\pi\)

\(\Leftrightarrow\dfrac{-25\pi}{12}\le k\pi\le-\dfrac{13\pi}{12}\)

\(\Leftrightarrow-\dfrac{25}{12}\le k\le-\dfrac{13}{12}\)

\(\Leftrightarrow-6.5\approx-\dfrac{25}{12}\le k\le-\dfrac{13}{12}\approx-3.4\)

Do k ∈ Z nên k = -1

Vậy PT có 1 nghiệm / \(\left[-\pi;-2\pi\right]\)

Ta có: $sin(\frac{\pi}{6})=\frac{1}{2}$

Do đó $sin(\frac{\pi}{6})=sin(x+ \frac{\pi}{3})\Leftrightarrow \left[\begin{matrix} \frac{\pi}{6}=x+\frac{\pi}{3}+2k\pi & \\ \frac{\pi}{6}= \pi-x-\frac{\pi}{3}+2k\pi& \end{matrix}\right.,k\in\mathbb{Z}$

$\Leftrightarrow \left[\begin{matrix} x=-\frac{\pi}{6}-2k\pi& \\ x=\frac{\pi}{2}+2k\pi& \end{matrix}\right.k\in\mathbb{Z}$

Vì $x \in [-\pi;-2\pi]$ nên ta có:

$\left[\begin{matrix} -\pi\ge \frac{-\pi}{6}-2k\pi\ge-2\pi & \\ -\pi\ge \frac{\pi}{2}+2k\pi\ge-2\pi \end{matrix}\right.\Leftrightarrow \left[\begin{matrix} -\frac{5\pi}{6}\ge -2k\pi\ge-\frac{11\pi}{6} & \\ -\frac{3\pi}{2}\ge +2k\pi\ge-\frac{5\pi}{2} \end{matrix}\right.\Leftrightarrow \left[\begin{matrix} \frac{5}{12}\le k\le \frac{11}{12} & \\ -\frac{3}{4}\ge k \ge-\frac{5}{4} & \end{matrix}\right.$

Vì $k\in\mathbb{Z}$ nên: 

$k=-1$

Vậy phương trình có 1 nghiệm trên $[-\pi;-2\pi]$

P/s: em mới học lớp 10 nên không biết làm thế này có đúng không ạ

Ta có: $sin(\frac{\pi}{6})=\frac{1}{2}$

Do đó $sin(\frac{\pi}{6})=sin(x+ \frac{\pi}{3})\Leftrightarrow \left[\begin{matrix} \frac{\pi}{6}=x+\frac{\pi}{3}+2k\pi & \\ \frac{\pi}{6}= \pi-x-\frac{\pi}{3}+2k\pi& \end{matrix}\right.,k\in\mathbb{Z}$

$\Leftrightarrow \left[\begin{matrix} x=-\frac{\pi}{6}-2k\pi& \\ x=\frac{\pi}{2}+2k\pi& \end{matrix}\right.k\in\mathbb{Z}$

Vì $x \in [-\pi;-2\pi]$ nên ta có:

$\left[\begin{matrix} -\pi\ge \frac{-\pi}{6}-2k\pi\ge-2\pi & \\ -\pi\ge \frac{\pi}{2}+2k\pi\ge-2\pi \end{matrix}\right.\Leftrightarrow \left[\begin{matrix} -\frac{5\pi}{6}\ge -2k\pi\ge-\frac{11\pi}{6} & \\ -\frac{3\pi}{2}\ge +2k\pi\ge-\frac{5\pi}{2} \end{matrix}\right.\Leftrightarrow \left[\begin{matrix} \frac{5}{12}\le k\le \frac{11}{12} & \\ -\frac{3}{4}\ge k \ge-\frac{5}{4} & \end{matrix}\right.$

Vì $k\in\mathbb{Z}$ nên: 

$k=-1$

Vậy phương trình có 1 nghiệm trên $[-\pi;-2\pi]$

`1)`

$a\big)\Delta=7^2-5.4.1=29>0\to$ PT có 2 nghiệm pb

$b\big)$

Theo Vi-ét: \(\left\{{}\begin{matrix}x_1+x_2=\dfrac{7}{5}\\x_1x_2=\dfrac{1}{5}\end{matrix}\right.\)

\(A=\left(x_1-\dfrac{7}{5}\right)x_1+\dfrac{1}{25x_2^2}+x_2^2\\ \Rightarrow A=\left(x_1-x_1-x_2\right)x_1+\left(\dfrac{1}{5}\right)^2\cdot\dfrac{1}{x_2^2}+x_2^2\\ \Rightarrow A=-x_1x_2+\left(x_1x_2\right)^2\cdot\dfrac{1}{x_2^2}+x_2^2\)

\(\Rightarrow A=-x_1x_2+x_1^2+x_2^2\\ \Rightarrow A=\left(x_1+x_2\right)^2-3x_1x_2\\ \Rightarrow A=\left(\dfrac{7}{5}\right)^2-3\cdot\dfrac{1}{5}=\dfrac{34}{25}\)

h) \(\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}=2\\\dfrac{3}{x}-\dfrac{4}{y}=-1\end{matrix}\right.\)\(\left(1\right)\)\(\left(đk:x,y\ne0\right)\)

Đặt \(a=\dfrac{1}{x},b=\dfrac{1}{y}\)

\(\left(1\right)\Leftrightarrow\) \(\left\{{}\begin{matrix}a+b=2\\3a-4b=-1\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}3a+3b=6\\3a-4b=-1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=2\\7b=7\end{matrix}\right.\)\(\Leftrightarrow a=b=1\)

Thay a,b:

\(\Leftrightarrow\dfrac{1}{x}=\dfrac{1}{y}=1\Leftrightarrow x=y=1\left(tm\right)\)