Bài 6:

a: ĐKXĐ: x∉{0;2}

Ta có: \(\frac{1}{x}+\frac{2}{x\left(x-2\right)}=\frac{x+2}{x-2}\)

=>\(\frac{x-2}{x\left(x-2\right)}+\frac{2}{x\left(x-2\right)}=\frac{x\left(x+2\right)}{x\left(x-2\right)}\)

=>\(x-2+2=x\left(x+2\right)\)

=>x(x+2)=x

=>x(x+2)-x=0

=>x(x+2-1)=0

=>x(x+1)=0

=>\(\left[\begin{array}{l}x=0\left(loại\right)\\ x+1=0\end{array}\right.\Rightarrow x+1=0\)

=>x=-1(nhận )

b: ĐKXĐ: y∉{0;-5;5}

Ta có: \(\frac{y+5}{y^2-5y}-\frac{y-5}{2y^2+10y}=\frac{y+25}{2y^2-50}\)

=>\(\frac{y+5}{y\left(y-5\right)}-\frac{y-5}{2y\left(y+5\right)}=\frac{y+25}{2\left(y-5\right)\left(y+5\right)}\)

=>\(\frac{2\left(y+5\right)^2}{2y\left(y+5\right)\left(y-5\right)}-\frac{\left(y-5\right)^2}{2y\left(y+5\right)\left(y-5\right)}=\frac{y\left(y+25\right)}{2y\left(y+5\right)\left(y-5\right)}\)

=>\(2\left(y+5\right)^2-\left(y-5\right)^2=y\left(y+25\right)\)

=>\(2y^2+20y+50-y^2+10y-25=y^2+25y\)

=>\(y^2+30y+25=y^2+25y\)

=>5y=-25

=>y=-5(loại)

Bài 7:

a: ĐKXĐ: x<>1

\(\frac{1}{x-1}+\frac{2x^2-5}{x^3-1}=\frac{4}{x^2+x+1}\)

=>\(\frac{1}{x-1}+\frac{2x^2-5}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{4}{x^2+x+1}\)

=>\(\frac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{2x^2-5}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{4\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)

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

=>\(3x^2+x-4=4x-4\)

=>\(3x^2-3x=0\)

=>3x(x-1)=0

=>x(x-1)=0

=>\(\left[\begin{array}{l}x=0\left(nhận\right)\\ x=1\left(loại\right)\end{array}\right.\)

b: ĐKXĐ: x<>2

Ta có: \(\frac{2x^2}{x^3-8}+\frac{x+1}{x^2+2x+4}=\frac{3}{x-2}\)

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

=>\(\frac{2x^2}{\left(x-2\right)\cdot\left(x^2+2x+4\right)}+\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x^2+2x+4\right)}=\frac{3\left(x^2+2x+4\right)}{\left(x-2\right)\left(x^2+2x+4\right)}\)

=>\(2x^2+\left(x+1\right)\left(x-2\right)=3\left(x^2+2x+4\right)\)

=>\(2x^2+x^2-x-2=3x^2+6x+12\)

=>6x+12=-x-2

=>7x=-14

=>x=-2(nhận)

c: ĐKXĐ: x∉{1;4}

Ta có: \(\frac{2x+1}{x^2-5x+4}+\frac{5}{x-1}=\frac{2}{x-4}\)

=>\(\frac{2x+1}{\left(x-1\right)\left(x-4\right)}+\frac{5}{x-1}=\frac{2}{x-4}\)

=>\(\frac{2x+1}{\left(x-1\right)\left(x-4\right)}+\frac{5\left(x-4\right)}{\left(x-1\right)\left(x-4\right)}=\frac{2\left(x-1\right)}{\left(x-1\right)\left(x-4\right)}\)

=>2x+1+5(x-4)=2(x-1)

=>2x+1+5x-20=2x-2

=>7x-19=2x-2

=>5x=17

=>\(x=\frac{17}{5}\) (nhận)

Bài 1:

a: \(\left(x-4\right)^3=\left(x+4\right)\left(x^2-x-16\right)\)

=>\(x^3-12x^2+48x-64=x^3-x^2-16x+4x^2-4x-64\)

=>\(x^3-12x^2+48x-64=x^3+3x^2-20x-64\)

=>\(-15x^2+68x=0\)

=>x(-15x+68)=0

=>\(\left[\begin{array}{l}x=0\\ -15x+68=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=\frac{68}{15}\end{array}\right.\)

b: ĐKXĐ: x∉{0;-2}

Ta có: \(\frac{x+2}{x}=\frac{x^2+5x+4}{x^2+2x}+\frac{x}{x+2}\)

=>\(\frac{x+2}{x}=\frac{x^2+5x+4}{x\left(x+2\right)}+\frac{x}{x+2}\)

=>\(\frac{\left(x+2\right)^2}{x\left(x+2\right)}=\frac{x^2+5x+4}{x\left(x+2\right)}+\frac{x^2}{x\left(x+2\right)}\)

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

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

=>\(x^2+x=0\)

=>x(x+1)=0

=>\(\left[\begin{array}{l}x=0\\ x+1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\left(loại\right)\\ x=-1\left(nhận\right)\end{array}\right.\)

c: ĐKXĐ: x∉{2;-2}

Ta có: \(\frac{x+1}{x-2}-\frac{5}{x+2}=\frac{12}{x^2-4}+1\)

=>\(\frac{\left(x+1\right)}{x-2}-\frac{5}{x+2}=\frac{12}{\left(x-2\right)\left(x+2\right)}-1\)

=>\(\frac{\left(x+1\right)\left(x+2\right)-5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}=\frac{12-\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}\)

=>\(\left(x+1\right)\left(x+2\right)-5\left(x-2\right)=12-\left(x-2\right)\left(x+2\right)\)

=>\(x^2+3x+2-5x+10=12-\left(x^2-4\right)\)

=>\(x^2-2x+12=12-x^2+4\)

=>\(x^2-2x+12=-x^2+16\)

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

=>\(x^2-x-2=0\)

=>(x-2)(x+1)=0

=>\(\left[\begin{array}{l}x-2=0\\ x+1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=2\left(loại\right)\\ x=-1\left(nhận\right)\end{array}\right.\)

Bài 2:

Gọi số học sinh giỏi là x(bạn)

(Điều kiện: x∈N*)

Số học sinh khá là \(\frac52x\) (bạn)

Số học sinh giỏi sau khi thêm 10 bạn là x+10(bạn)

Số học sinh khá sau khi bớt đi 6 bạn là \(\frac52x-6\) (bạn)

Số học sinh khá sẽ gấp 2 lần số học sinh giỏi nên ta có:

\(\frac52x-6=2\left(x+10\right)\)

=>2,5x-6=2x+20

=>0,5x=26

=>x=52(nhận)

vậy: Số học sinh giỏi là 52 bạn

Bài 2: Để hệ có nghiệm duy nhất thì \(\frac{1}{a}<>\frac{a}{1}\)

=>\(a^2<>1\)

=>a∉{1;-1](1)

\(\begin{cases}ax+y=3a\\ x+ay=2a+1\end{cases}\Rightarrow\begin{cases}y=3a-ax\\ x+a\left(3a-ax\right)=2a+1\end{cases}\)

=>\(\begin{cases}y=3a-a\cdot x\\ x+3a^2-a^2\cdot x=2a+1\end{cases}\Rightarrow\begin{cases}y=3a-ax\\ x\left(1-a^2\right)=2a+1-3a^2\end{cases}\)

=>\(\begin{cases}x=\frac{-3a^2+2a+1}{1-a^2}=\frac{3a^2-2a-1}{a^2-1}=\frac{\left(a-1\right)\left(3a+1\right)}{\left(a-1\right)\left(a+1\right)}=\frac{3a+1}{a+1}\\ y=3a-a\cdot\frac{3a+1}{a+1}=\frac{3a^2+3a-3a^2-a}{a+1}=\frac{2a}{a+1}\end{cases}\)

Để x,y nguyên thì \(\begin{cases}3a+1\vdots a+1\\ 2a\vdots a+1\end{cases}\Rightarrow\begin{cases}3a+3-2\vdots a+1\\ 2a+2-2\vdots a+1\end{cases}\)

=>-2⋮a+1

=>a+1∈{1;-1;2;-2}

=>a∈{0;-2;1;-3}

Kết hợp (1), ta có: a∈{0;-2;-3}

Bài 3:

ĐKXĐ: x>=y

\(\begin{cases}\sqrt{\frac{x+y}{2}}+\sqrt{\frac{x-y}{3}}=14\\ \sqrt{\frac{x+y}{8}}-\sqrt{\frac{x-y}{12}}=3\end{cases}\Rightarrow\begin{cases}\sqrt{\frac{x+y}{2}}+\sqrt{\frac{x-y}{3}}=14\\ \frac12\left(\sqrt{\frac{x+y}{2}}-\sqrt{\frac{x-y}{3}}\right)=3\end{cases}\)

=>\(\begin{cases}\sqrt{\frac{x+y}{2}}+\sqrt{\frac{x-y}{3}}=14\\ \sqrt{\frac{x+y}{2}}-\sqrt{\frac{x-y}{3}}=6\end{cases}\Rightarrow\begin{cases}\sqrt{\frac{x+y}{2}}=10\\ \sqrt{\frac{x-y}{3}}=4\end{cases}\)

=>\(\begin{cases}\frac{x+y}{2}=100\\ \frac{x-y}{3}=16\end{cases}\Rightarrow\begin{cases}x+y=200\\ x-y=48\end{cases}\Rightarrow\begin{cases}x=\frac{200+48}{2}=\frac{248}{2}=124\\ y=200-124=76\end{cases}\) (nhận)

Bài 3:

a: \(\left(2x+1\right)\left(x^2+2\right)=0\)

mà \(x^2+2\ge2>0\forall x\)

nên 2x+1=0

=>2x=-1

=>\(x=-\frac12\)

b: \(\left(x^2+4\right)\left(7x-3\right)=0\)

mà \(x^2+4\ge4>0\forall x\)

nên 7x-3=0

=>7x=3

=>\(x=\frac37\)

c: \(\left(x^2+x+1\right)\left(6-2x\right)=0\)

mà \(x^2+x+1=x^2+x+\frac14+\frac34=\left(x+\frac12\right)^2+\frac34\ge\frac34>0\forall x\)

nên 6-2x=0

=>2x=6

=>x=3

d: \(\left(8x-4\right)\left(x^2+2x+2\right)=0\)

mà \(x^2+2x+2=x^2+2x+1+1=\left(x+1\right)^2+1\ge1>0\forall x\)

nên 8x-4=0

=>8x=4

=>\(x=\frac48=\frac12\)

Bài 4:

a: \(\left(x-2\right)\left(3x+5\right)=\left(2x-4\right)\left(x+1\right)\)

=>(x-2)(3x+5)=(x-2)(2x+2)

=>(x-2)(3x+5-2x-2)=0

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

=>\(\left[\begin{array}{l}x-2=0\\ x+3=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=2\\ x=-3\end{array}\right.\)

b: \(\left(2x+5\right)\left(x-4\right)=\left(x-5\right)\left(4-x\right)\)

=>(2x+5)(x-4)-(x-5)(4-x)=0

=>(2x+5)(x-4)+(x-5)(x-4)=0

=>(x-4)(2x+5+x-5)=0

=>3x(x-4)=0

=>x(x-4)=0

=>\(\left[\begin{array}{l}x=0\\ x-4=0\end{array}\right.=>\left[\begin{array}{l}x=0\\ x=4\end{array}\right.\)

c: \(9x^2-1=\left(3x+1\right)\left(2x-3\right)\)

=>(3x+1)(3x-1)=(3x+1)(2x-3)

=>(3x+1)(3x-1)-(3x+1)(2x-3)=0

=>(3x+1)(3x-1-2x+3)=0

=>(3x+1)(x+2)=0

=>\(\left[\begin{array}{l}3x+1=0\\ x+2=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-\frac13\\ x=-2\end{array}\right.\)

d: \(2\left(9x^2+6x+1\right)=\left(3x+1\right)\left(x-2\right)\)

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

=>\(\left(3x+1\right)\left(6x+2-x+2\right)=0\)

=>(3x+1)(5x+4)=0

=>\(\left[\begin{array}{l}3x+1=0\\ 5x+4=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-\frac13\\ x=-\frac45\end{array}\right.\)

e: \(27x^2\left(x+3\right)-12\left(x^2+3x\right)=0\)

=>\(27x^2\left(x+3\right)-12x\left(x+3\right)=0\)

=>3x(x+3)(9x-4)=0

=>x(x+3)(9x-4)=0

=>\(\left[\begin{array}{l}x=0\\ x+3=0\\ 9x-4=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=-3\\ x=\frac49\end{array}\right.\)

f: \(16x^2-8x+1=4\left(x+3\right)\left(4x-1\right)\)

=>\(\left(4x-1\right)^2=\left(4x+12\right)\left(4x-1\right)\)

=>(4x+12)(4x-1)-\(\left(4x-1\right)^2=0\)

=>(4x-1)(4x+12-4x+1)=0

=>13(4x-1)=0

=>4x-1=0

=>4x=1

=>\(x=\frac14\)

ĐKXĐ: x∉{2;-1;-2}

Ta có: \(\frac{3}{x^2-x-2}+\frac{3}{x^2+3x+2}=\frac{3}{x^2+4}\)

=>\(\frac{1}{x^2-x-2}+\frac{1}{x^2+3x+2}=\frac{1}{x^2+4}\)

=>\(\frac{1}{\left(x-2\right)\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}=\frac{1}{x^2+4}\)

=>\(\frac{x+2+x-2}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}=\frac{1}{x^2+4}\)

=>\(\frac{2x}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}=\frac{1}{x^2+4}\)

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

=>\(2x^3+8x=x^3-4x-x^2+4\)

=>\(x^3+x^2+12x-4=0\)

=>x≃0,32(nhận)