Phương trình có nghiệm

Ai chẳng biết bạn

Ta có: m(m - 1)x = m² + 3m + 2(x + 1)

=> (m² - m)x = m² + 3m + 2x + 2

=> (m² - m - 2)x = m² + 3m + 2

Để pt có nghiệm duy nhất thì m² - m - 2 \(\ne0\Rightarrow m\ne-1\) và \(m\ne2\)

\(\Rightarrow x=\frac{m^2-m-2}{m^2+3m+2}=\frac{\left(m+1\right)\left(m-2\right)}{\left(m+1\right)\left(m+2\right)}=\frac{m-2}{m+2}\)

Theo đề: \(\frac{m-2}{m+2}<1\Rightarrow\frac{m-2}{m+2}-1<0\Rightarrow\frac{-4}{m+2}<0\)

=> m + 2 > 0 => m > -2

Vậy m > -2 và m\(\ne-1,m\ne2\)

\(\left(m-1\right)x=2-3m\) (với \(m\ne1\))

\(\Rightarrow x=\dfrac{2-3m}{m-1}\)

\(x\ge1\Rightarrow\dfrac{2-3m}{m-1}\ge1\)

\(\Rightarrow\dfrac{2-3m}{m-1}-1\ge0\Rightarrow\dfrac{3-4m}{m-1}\ge0\)

\(\Rightarrow\dfrac{3}{4}\le m< 1\)

\( (m-1)x+3m-2 =0 \\ \Leftrightarrow x= \dfrac{2-3m}{m-1} \\ \Rightarrow \) PT có nghiệm \(\Leftrightarrow m-1 \ne 0 \Leftrightarrow m \ne 1\)

\(x ≥ 1 \Leftrightarrow 2-3m ≥ m-1 \Leftrightarrow m ≤ \dfrac{3}{4}\)

Vậy \(m ≤ \dfrac{3}{4}\).

PT có nghiệm duy nhất khi và chỉ khi m - 1 khác 0, tức m khác 1.

Khi đó \(x=\dfrac{2-3m}{m-1}\).

\(x\ge1\Leftrightarrow\dfrac{2-3m}{m-1}\ge1\Leftrightarrow\dfrac{2-3m-m+1}{m-1}\ge0\Leftrightarrow\dfrac{3-4m}{m-1}\ge0\Leftrightarrow\dfrac{4}{3}\ge m>1\).

Vậy ....

a) Ta có: \(x^2+\dfrac{9x^2}{\left(x+3\right)^2}=40\)

\(\Leftrightarrow\dfrac{\left(x^2+3x\right)^2+9x^2}{\left(x+3\right)^2}=40\)

\(\Leftrightarrow x^4+6x^3+9x^2+9x^2=40\left(x+3\right)^2\)

\(\Leftrightarrow x^4+6x^3+18x^2=40\left(x^2+6x+9\right)\)

\(\Leftrightarrow x^4+6x^3+18x^2-40x^2-240x-360=0\)

\(\Leftrightarrow x^4+6x^3-22x^2-240x-360=0\)

\(\Leftrightarrow x^4+2x^3+4x^3+8x^2-30x^2-60x-180x-360=0\)

\(\Leftrightarrow x^3\left(x+2\right)+4x^2\left(x+2\right)-30x\left(x+2\right)-180\left(x+2\right)=0\)

\(\Leftrightarrow\left(x+2\right)\left(x^3+4x^2-30x-180\right)=0\)

\(\Leftrightarrow\left(x+2\right)\left(x^3-6x^2+10x^2-60x+30x-180\right)=0\)

\(\Leftrightarrow\left(x+2\right)\left[x^2\left(x-6\right)+10x\left(x-6\right)+30\left(x-6\right)\right]=0\)

\(\Leftrightarrow\left(x+2\right)\cdot\left(x-6\right)\left(x^2+10x+30\right)=0\)

mà \(x^2+10x+30>0\forall x\)

nên \(\left(x+2\right)\left(x-6\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x+2=0\\x-6=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-2\\x=6\end{matrix}\right.\)

Vậy: S={-2;6}

b) Ta có: (m-1)x+3m-2=0

\(\Leftrightarrow\left(m-1\right)x=2-3m\)

\(\Leftrightarrow x=\dfrac{2-3m}{m-1}\)

Để phương trình có nghiệm duy nhất thỏa mãn \(x\ge1\) thì \(\dfrac{2-3m}{m-1}\ge1\)

\(\Leftrightarrow\dfrac{2-3m}{m-1}-1\ge0\)

\(\Leftrightarrow\dfrac{2-3m-\left(m-1\right)}{m-1}\ge0\)

\(\Leftrightarrow\dfrac{2-3m-m+1}{m-1}\ge0\)

\(\Leftrightarrow\dfrac{-4m+3}{m-1}\ge0\)

hay \(\dfrac{3}{4}\le m< 1\)

Vậy: Để phương trình (m-1)x+3m-2=0 có nghiệm duy nhất thỏa mãn \(x\ge1\) thì \(\dfrac{3}{4}\le m< 1\)

a: Thay m=-2 vào hệ phương trình, ta được:

\(\left\{{}\begin{matrix}x-2y=-2+1=-1\\-2x+y=3\cdot\left(-2\right)-1=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2x-4y=-2\\-2x+y=-7\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-3y=-9\\x-2y=-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=3\\x=2y-1=2\cdot3-1=5\end{matrix}\right.\)

b: Để hệ có nghiệm duy nhất thì \(\dfrac{1}{m}\ne\dfrac{m}{1}\)

=>\(m^2\ne1\)

=>\(m\notin\left\{1;-1\right\}\)

\(\left\{{}\begin{matrix}x+my=m+1\\mx+y=3m-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=m+1-my\\m\left(m+1-my\right)+y=3m-1\end{matrix}\right.\)

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

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

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

=>\(\left\{{}\begin{matrix}y=\dfrac{m-1}{m+1}\\x=m+1-m\cdot\dfrac{m-1}{m+1}=\left(m+1\right)-\dfrac{m^2-m}{m+1}\end{matrix}\right.\)

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

\(x^2-y^2=4\)

=>\(\dfrac{\left(3m+1\right)^2-\left(m-1\right)^2}{\left(m+1\right)^2}=4\)

=>\(\dfrac{9m^2+6m+1-m^2+2m+1}{\left(m+1\right)^2}=4\)

=>\(8m^2+8m+2=4\left(m+1\right)^2\)

=>\(8m^2+8m+2-4m^2-8m-4=0\)

=>\(4m^2-2=0\)

=>\(m^2=\dfrac{1}{2}\)

=>\(m=\pm\dfrac{1}{\sqrt{2}}\)