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

\(=m^2-8m+16+4m-12\)

\(=m^2-4m+4=\left(m-2\right)^2>=0\)

Do đó: Phương trình luôn có hai nghiệm với mọi m

Theo đề, ta có hệ phương trình:

\(\left\{{}\begin{matrix}3x_1-x_2=2\\x_1+x_2=-m+4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4x_1=6-m\\x_2=3x_1-2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x_1=\dfrac{6-m}{4}\\x_2=\dfrac{3\left(6-m\right)}{4}-2=\dfrac{18-3m-8}{4}=\dfrac{10-3m}{4}\end{matrix}\right.\)

Theo đề, ta có: \(x_1x_2=-m+3\)

\(\Leftrightarrow\left(m-6\right)\left(3m-10\right)=16\left(-m+3\right)\)

\(\Leftrightarrow3m^2-30m-18m+60+16m-48=0\)

\(\Leftrightarrow3m^2-32m+12=0\)

\(\text{Δ}=\left(-32\right)^2-4\cdot3\cdot12=880>0\)

Do đó: Phương trình có hai nghiệm phân biệt là:

\(\left\{{}\begin{matrix}x_1=\dfrac{32-4\sqrt{55}}{6}=\dfrac{16-2\sqrt{55}}{3}\\x_2=\dfrac{16+2\sqrt{55}}{3}\end{matrix}\right.\)

can tui giup k

\(a,\left(x-6\right)\left(2x-5\right)\left(3x+9\right)=0\Leftrightarrow\left[{}\begin{matrix}x-6=0\Leftrightarrow x=6\\2x-5=0\Leftrightarrow x=\dfrac{5}{2}\\3x+9=0\Leftrightarrow x=-3\end{matrix}\right.\)

\(b,2x\left(x-3\right)+5\left(x-3\right)=0\Leftrightarrow\left(2x+5\right)\left(x-3\right)=0\Leftrightarrow\left[{}\begin{matrix}x-3=0\Leftrightarrow x=3\\2x+5=0\Leftrightarrow x=-\dfrac{5}{2}\end{matrix}\right.\)

\(c,x^2-4-\left(x-2\right)\left(3-2x\right)=0\Leftrightarrow\left(x-2\right)\left(x+2\right)-\left(x-2\right)\left(3-2x\right)=0\Leftrightarrow\left(x-2\right)\left(x+2-3+2x\right)=0\Leftrightarrow\left[{}\begin{matrix}x=2\\x=\dfrac{1}{3}\end{matrix}\right.\)

\(x=-7\left(2m-5\right)x-2m^2+8\Leftrightarrow x+7\left(2m-5\right)=8-2m^2\Leftrightarrow x\left(14m-34\right)=8-2m^2\)

\(ycđb\Leftrightarrow14m-34\ne0\Leftrightarrow m\ne\dfrac{34}{14}\)\(\Rightarrow x=\dfrac{8-2m^2}{14m-34}\)

\(3.17\Leftrightarrow4x^2-4x+1-2x-1=0\Leftrightarrow4x^2-6x=0\Leftrightarrow x\left(4x-6\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{3}{2}\end{matrix}\right.\)

3.15:

a, \(\Leftrightarrow\left\{{}\begin{matrix}x-6=0\\2x-5=0\\3x+9=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=6\\x=\dfrac{5}{2}\\x=-\dfrac{9}{3}=-3\end{matrix}\right.\)

b, \(\Leftrightarrow\left(x-3\right)\left(2x+5\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-3=0\\2x+5=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\x=-\dfrac{5}{2}\end{matrix}\right.\)

c, \(\Leftrightarrow\left(x-2\right)\left(x+2\right)-\left(x-2\right)\left(3-2x\right)=0\)

\(\Leftrightarrow\left(x-2\right)\left(x+2-3+2x\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\3x-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\x=\dfrac{1}{3}\end{matrix}\right.\)

3.16

\(\Leftrightarrow\left(2m-5\right).-7-2m^2+8=0\)

\(\Leftrightarrow-14m+35-2m^2+8=0\)

\(\Leftrightarrow-14m-2m^2+43=0\)

\(\Leftrightarrow-2\left(7m+m^2\right)=-43\)

\(\Leftrightarrow m\left(7-m\right)=\dfrac{43}{2}\)

\(\Leftrightarrow\dfrac{m\left(7-m\right)}{1}-\dfrac{43}{2}=0\)

\(\Leftrightarrow\dfrac{14m-2m^2}{2}-\dfrac{43}{2}=0\)

pt vô nghiệm

- Với \(m=2\) pt có nghiệm

- Với \(m\ne2\) để pt có nghiệm

\(\Rightarrow\Delta'=\left(2m-3\right)^2-\left(m-2\right)\left(5m-6\right)\ge0\)

\(\Leftrightarrow-m^2+4m-3\ge0\Rightarrow1\le m\le3\)

Vậy \(1\le m\le3\)

\(4sin\left(x+\dfrac{\pi}{3}\right).cos\left(x-\dfrac{\pi}{6}\right)=m^2+\sqrt[]{3}sin2x-cos2x\)

\(\Leftrightarrow4.\left(-\dfrac{1}{2}\right)\left[sin\left(x+\dfrac{\pi}{3}+x-\dfrac{\pi}{6}\right)+sin\left(x+\dfrac{\pi}{3}-x+\dfrac{\pi}{6}\right)\right]=m^2+2.\left[\dfrac{\sqrt[]{3}}{2}.sin2x-\dfrac{1}{2}.cos2x\right]\)

\(\Leftrightarrow2\left[sin\left(2x+\dfrac{\pi}{6}\right)+sin\left(2x-\dfrac{\pi}{6}\right)\right]=m^2+2\)

\(\Leftrightarrow2.2sin2x.cos\dfrac{\pi}{6}=m^2+2\)

\(\Leftrightarrow2.2sin2x.\dfrac{\sqrt[]{3}}{2}=m^2+2\)

\(\Leftrightarrow2\sqrt[]{3}sin2x.=m^2+2\)

\(\Leftrightarrow sin2x.=\dfrac{m^2+2}{2\sqrt[]{3}}\)

Phương trình có nghiệm khi và chỉ khi

\(\left|\dfrac{m^2+2}{2\sqrt[]{3}}\right|\le1\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{m^2+2}{2\sqrt[]{3}}\ge-1\\\dfrac{m^2+2}{2\sqrt[]{3}}\le1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m^2\ge-2\left(1+\sqrt[]{3}\right)\left(luôn.đúng\right)\\m^2\le2\left(1-\sqrt[]{3}\right)\end{matrix}\right.\)

\(\Leftrightarrow-\sqrt[]{2\left(1-\sqrt[]{3}\right)}\le m\le\sqrt[]{2\left(1-\sqrt[]{3}\right)}\)

Bài 5 :

Thay \(x=-3\) vào pt : \(3x+m-x-1=0\)

\(\Leftrightarrow3\left(-3\right)+m-\left(-3\right)-1=0\)

\(\Leftrightarrow-9+m+3-1=0\)

\(\Leftrightarrow m-7=0\)

\(\Leftrightarrow m=7\)

Vậy \(m=7\) để pt nhận \(x=-3\) là nghiệm

Bài 6 :

Thay \(x=1\) vào pt : \(\left(2m-4\right)x+6=0\)

\(\Leftrightarrow2mx-4x+6=0\)

\(\Leftrightarrow2m-4+6=0\)

\(\Leftrightarrow2m+2=0\)

\(\Leftrightarrow m=-1\)

Vậy \(m=-1\) để pt nhận \(x=1\) là nghiệm