\(\left(12x+7\right)^2\left(3x+2\right)\left(2x+1\right)=3\)

\(\Leftrightarrow\left(12x+7\right)^2\cdot4\left(3x+2\right)\cdot6\left(2x+1\right)=3\cdot4\cdot6\)

\(\Leftrightarrow\left(12x+7\right)^2\left(12x+8\right)\left(12x+6\right)=72\) (1)

Đặt 12x + 7 = a

(1) \(\Leftrightarrow a^2\left(a+1\right)\left(a-1\right)=72\)

\(\Leftrightarrow a^2\left(a^2-1\right)=72\) (2)

Đặt \(a^2=b\)

(2) \(\Leftrightarrow b\left(b-1\right)=72\)

\(\Leftrightarrow b^2-b-72=0\)

\(\Leftrightarrow b^2+8b-9b-72=0\)

\(\Leftrightarrow b\left(b+8\right)-9\left(b+8\right)=0\)

\(\Leftrightarrow\left(b-9\right)\left(b+8\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}b-9=0\\b+8=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}b=9\\b=-8\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}a^2=9\Leftrightarrow a=\pm3\\a^2=-8\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}12x+7=3\\12x+7=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}12x=-4\\12x=-10\end{matrix}\right.\Leftrightarrow}\left[{}\begin{matrix}x=-\dfrac{1}{3}\\x=-\dfrac{5}{6}\end{matrix}\right.\)

\(a^2\)không thể bằng -8 nên hãy xem xét lại

Lời giải:

Tập xác định của phương trình

Biến đổi vế trái của phương trình

Phương trình thu được sau khi biến đổi

\(\Leftrightarrow\left(144x^2+168x+49\right)\left(6x^2+7x+2\right)=3\)

Đặt \(6x^2+7x+2=t\Rightarrow6x^2+7x=t-2\)

\(\Rightarrow144x^2+168x+49=24\left(6x^2+7x\right)+49=24\left(t-2\right)+49=24t+1\)

Phương trình trở thành:

\(\left(24t+1\right)t=3\Leftrightarrow24t^2+t-3=0\Rightarrow\left[{}\begin{matrix}t=\dfrac{1}{3}\\t=-\dfrac{3}{8}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}6x^2+7x+2=\dfrac{1}{3}\\6x^2+7x+2=-\dfrac{3}{8}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}6x^2+7x+\dfrac{5}{3}=0\\6x^2+7x+\dfrac{19}{8}=0\end{matrix}\right.\) (bấm máy)

\(\left(12x+7\right)^2\left(3x+2\right)\left(2x+1\right)=3\)

\(\Leftrightarrow\left(144x^2+168x+49\right)\left(6x^2+7x+2\right)=3\)

\(\Leftrightarrow\left(144x^2+168x+49\right)\left(144x^2+168+48\right)=72\)

Đặt \(144x^2+168x+48=u\)

\(\Rightarrow144x^2+168x+49=u+1\left(1\right)\)

Do đó: \(u\left(u+1\right)=72\Leftrightarrow u^2+u-72=0\)

\(\Leftrightarrow\left(u-8\right)\left(u+9\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}u-8=0\\u+9=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}u=8\\u=-9\end{matrix}\right.\)

Với \(u=8;u=-9\) bạn thay vào (1) và tìm x nha.

f. 5 – (x – 6) = 4(3 – 2x)

<=>5-x+6=12-8x

<=>7x=1

<=>x=\(\dfrac{1}{7}\)

g. 7 – (2x + 4) = – (x + 4)

<=>7-2x-4=-x-4

<=>x=7

h. 

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

<=>\(2x^3+8x^2+8x-8x^2=2\left(x^3-8\right)\)

<=>\(2x^3+8x=2x^3-16\)

<=>\(8x=-16\)

<=>\(x=-2\)

i. 

k. (x + 1)(2x – 3) = (2x – 1)(x + 5)

<=>\(2x^2-x-3=2x^2+9x-5\)

<=>10x=2

<=>\(x=\dfrac{1}{5}\)

f. 5 – (x – 6) = 4(3 – 2x)

<=>5-x+6=12-8x

<=>7x=1

<=>x=\(\dfrac{1}{7}\)

g. 7 – (2x + 4) = – (x + 4)

<=>7-2x-4=-x-4

<=>x=7

h. \(2x\left(x+2\right)^2-8x^2=2\left(x-2\right)\left(x^2+2x+4\right)\)

<=>

<=>

<=>\(8x=-16\)

<=>x=-2

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

<=>\(x^3-6x^2+12x+8+9x^2-1=x^3+3x^2+3x+1\)

<=>\(9x+6=0\)

<=>x=\(\dfrac{-2}{3}\)

k. (x + 1)(2x – 3) = (2x – 1)(x + 5)

<=>

<=>10x=2

<=>

a: =>x+3=x-2 hoặc x+3=2-x

=>2x=-1

=>x=-1/2

b: =>3x+7=x-2 hoặc 3x+7=-x+2

=>2x=-9 hoặc 4x=-5

=>x=-5/4 hoặc x=-9/2

c: =>|3x-4|=|2x-5|

=>3x-4=2x-5 hoặc 3x-4=-2x+5

=>x=-1 hoặc x=9/5

a)\(\frac{2-x}{2007}-1=\frac{1-x}{2008}-\frac{x}{2009}\)

\(\Leftrightarrow\frac{2-x}{2007}-1+2=\frac{1-x}{2008}+1-\frac{x}{2009}+1\)

\(\Leftrightarrow\frac{2-x}{2007}+\frac{2007}{2007}=\frac{1-x}{2008}+\frac{2008}{2008}-\frac{x}{2009}+\frac{2009}{2009}\)

\(\Leftrightarrow\frac{2009-x}{2007}=\frac{2009-x}{2008}-\frac{2009-x}{2009}\)

\(\Leftrightarrow\frac{2009-x}{2007}-\frac{2009-x}{2008}+\frac{2009-x}{2009}=0\)

\(\Leftrightarrow\left(2009-x\right)\left(\frac{1}{2007}-\frac{1}{2008}+\frac{1}{2009}\right)=0\)

\(\Leftrightarrow2009-x=0\).Do \(\frac{1}{2007}-\frac{1}{2008}+\frac{1}{2009}\ne0\)

\(\Leftrightarrow x=2009\)

b)\(\left(12x+7\right)^2\left(3x+2\right)\left(2x+1\right)=3\)

\(\Leftrightarrow\left(12^2x^2+2\cdot12\cdot7x+7^2\right)\left(6x^2+7x+2\right)-3=0\)

\(\Leftrightarrow\left[24\left(6x^2+7x+2\right)+1\right]\left(6x^2+7x+2\right)-3=0\)

Đặt \(t=6x^2+7x+2\) ta có:

\(\left(24t+1\right)t-3=0\)\(\Leftrightarrow12t^2+t-3=0\)

Suy ra t rồi tìm đc x

VD: 

INPUT: 4 

OUTPUT: 

1

1   1

1    2    1

1    3    3    1

1    4    6     4     1

a: =>|x-7|=3-2x

\(\Leftrightarrow\left\{{}\begin{matrix}x< =\dfrac{3}{2}\\\left(-2x+3\right)^2-\left(x-7\right)^2=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x< =\dfrac{3}{2}\\\left(2x-3-x+7\right)\left(2x-3+x-7\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x< =\dfrac{3}{2}\\\left(x+4\right)\left(3x-10\right)=0\end{matrix}\right.\Leftrightarrow x=-4\)

b: =>|2x-3|=4x+9

\(\Leftrightarrow\left\{{}\begin{matrix}x>=-\dfrac{9}{4}\\\left(4x+9-2x+3\right)\left(4x+9+2x-3\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x>=-\dfrac{9}{4}\\\left(2x+12\right)\left(6x+6\right)=0\end{matrix}\right.\Leftrightarrow x=-1\)

c: =>3x+5=2-5x hoặc 3x+5=5x-2

=>8x=-3 hoặc -2x=-7

=>x=-3/8 hoặc x=7/2