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\(\Leftrightarrow\left(2x-1\right)^3-\left(2x+3\right)^3-3\left(3x+1\right)^2-2\left(x-2\right)^2+\left(x-1\right)\left(x+2\right)=0\)
\(\Leftrightarrow8x^3-12x^2+6x-1-8x^3-36x^2-54x-27-3\left(9x^2+6x+1\right)-2\left(x^2-4x+4\right)+x^2+x-2=0\)
\(\Leftrightarrow-48x^2-48x-28-27x^2-18x-3-2x^2+8x-8+x^2+x-2=0\)
\(\Leftrightarrow-76x^2-57x-41=0\)
\(\Leftrightarrow76x^2+57x+41=0\)
\(\text{Δ}=57^2-4\cdot76\cdot41=-9215< 0\)
Vậy: Phương trình vô nghiệm
(x -2)\(^3\) +(3x-2)\(^2\) -5x (x+1) = (1+x)\(^3\) - 2(2x+1)\(^2\)
<=> (x\(^3\) -3.x\(^2\).2+3.x.2\(^2\) -2\(^3\)) + [(3x)\(^2\) - 2.3x.2 +2\(^2\)] - (5x.x+ 5x .1) = (1\(^3\) + 3.1\(^2\).x+ 3.1.x\(^2\) + x\(^3\) )- [2((2x)\(^2\) +2.2x.1+ 1\(^2\))]
<=> (x\(^3\) - 6x\(^2\) + 12x - 8) + (9x\(^2\) -12x+ 4)- (5x\(^2\) + 5x) = (1+3x + 3x\(^2\) + x\(^3\)) - [ 2.(4x\(^2\) + 4x +1]= (1+3x + 3x\(^2\) + x\(^3\)) - ( 8x\(^2\)+ 8x +2)
<=> x\(^3\) - 6x\(^2\) + 12x - 8 + 9x\(^2\) -12x+ 4 - 5x\(^2\) - 5x = 1+3x + 3x\(^2\) + x\(^3\) - 8x\(^2\) -8x - 2
<=> x\(^3\) +(- 6x\(^2\) + 9x\(^2\) - 5x\(^2\) ) +(12x- 12x - 5x) + (-8 +4) = (1-2) + ( 3x-8x) +( 3x\(^2\) - 8x\(^2\) ) + x\(^3\)
<=> x\(^3\) +( -2x\(^2\)) + (-5x) + (-4) = -1 + (-5x) +( -5x\(^2\))+ x\(^3\)<=> x\(^3\) -2x\(^2\) -5x-4= -1 - 5x - 5x\(^2\) +x\(^3\)<=> -2x\(^2\) -4 = -1 -5x\(^2\)<=> -2x\(^2\) + 5x\(^2\) = -1 +4 ( chuyển vế )<=> 3x\(^2\) = 3<=> x\(^2\) = 3:3<=> x\(^2\) = 1<=> x = \(\sqrt{1}\)<=> x= 1 CHÚC BẠN HỌC TỐT
\(\left(2x-1\right)^3-3\left(1-3x\right)^2=\left(3+2x\right)^3-2\left(x-2\right)\left(x+3\right)\)
\(8x^3-12x^2+6x-1-3\left(1-6x+9x^2\right)=27+54x+36x^2+8x^3-2\left(x^2+3x-2x-6\right)\)\(8x^3-12x^2+6x-1-3+18x-27x^2=27+54x+36x^2+8x^3-2x^2-6x+4x+12\)\(8x^3-39x^2+24x-4=8x^3+34x^2+52x+39\)
\(8x^3-39x^2+24x-4-8x^3-34x^2-52x-39=0\)
\(-73x^2-28x-43=0\)
Vậy đa thức vô nghiệm
\(\Leftrightarrow20\left(x^2-4x+3\right)-24\left(4x^2-4x+1\right)=15\left(9x^2+6x+1\right)+90x\left(x-1\right)\)
\(\Leftrightarrow20x^2-80x+60-96x^2+96x-24=135x^2+90x+15+90x^2-90x\)
\(\Leftrightarrow-301x^2+16x+21=0\)
\(\text{Δ}=16^2-4\cdot\left(-301\right)\cdot21=25540\)
Vì Δ>0 nên phương trình có hai nghiệm phân biệt là
\(\left\{{}\begin{matrix}x_1=\dfrac{-16-\sqrt{25540}}{-602}=\dfrac{16+\sqrt{25540}}{602}\\x_2=\dfrac{16-\sqrt{25540}}{602}\end{matrix}\right.\)
\(A=\left(\dfrac{x^2+1}{x^2\cdot\left(x+1\right)^2}+\dfrac{2}{\left(x+1\right)^3}\cdot\dfrac{x+1}{x}\right):\dfrac{x-1}{x^3}\)
\(=\dfrac{x^2+3}{x^2\cdot\left(x+1\right)^2}\cdot\dfrac{x^3}{x-1}=\dfrac{x\left(x^2+3\right)}{\left(x-1\right)\left(x+1\right)^2}\)
1. (3x - 5)2 - (3x + 1)2 = 8
=> (3x - 5 - 3x - 1)(3x - 5 + 3x + 1) = 8
=> -6(6x - 4) = 8
=> 6x - 4 = \(\dfrac{-4}{3}\)
\(\Rightarrow x=\dfrac{4}{9}\)
2) 2x(8x - 3) - (4x - 3)2 = 27
=> 16x2 - 6x - 16x2 + 24x - 9 = 27
=> 18x - 9 = 27
=> x = 2
3) (2x - 3)2 - (2x + 1)2 = 3
=> (2x - 3 - 2x - 1)(2x - 3 + 2x +1) = 3
=> -4(4x - 2) = 3
=> 4x - 2 = \(\dfrac{-3}{4}\)
\(\Rightarrow x=\dfrac{5}{16}\)
4) (x + 5)2 - x2 = 45
=> (x + 5 - x)(x + 5 + x) = 45
=> 5(2x + 5) = 45
=> 2x + 5 = 9
=> x = 2
5) (x - 3)3 - (x - 3)(x2 + 3x + 9) + 9(x + 1)2 = 18
=> x3 - 9x2 + 27x - 27 - x3 + 27 + 9(x2 + 2x + 1) = 18
=> -9x2 + 27x + 9x2 + 18x + 9 = 18
=> 45x + 9 = 18
=> 45x = 9
=> x = \(\dfrac{1}{5}\)
6) x(x - 4)(x + 4) - (x - 5)(x2 + 5x + 25) = 13
=> x (x2 - 16) - (x3 - 125) = 13
=> x3 - 16x - x3 + 125 = 13
=> -16x = -112
=> x = 7.
\(\left[\left(1+\frac{1}{x^2}\right)\div\left(1+2x+x^2\right)+\frac{2}{\left(x+1\right)^3}\times\left(1+\frac{1}{x}\right)\right]\div\frac{x-1}{x^3}\)
\(=\left[\frac{x^2+1}{x^2}\times\frac{1}{\left(x+1\right)^2}+\frac{2}{\left(x+1\right)^3}\times\frac{x+1}{x}\right]\div\frac{x-1}{x^3}\)
\(=\left(\frac{x^2+1}{x^2}\times\frac{1}{\left(x+1\right)^2}+\frac{1}{\left(x+1\right)^2}\times\frac{2}{x}\right)\div\frac{x-1}{x^3}\)
\(=\left(\frac{1}{\left(x+1\right)^2}\times\left(\frac{x^2+1}{x^2}+\frac{2}{x}\right)\right)\div\frac{x-1}{x^3}\)
\(=\left(\frac{1}{\left(x+1\right)^2}\times\frac{x^3+2x^2+x}{x^3}\right)\div\frac{x-1}{x^3}\)
\(=\left(\frac{1}{\left(x+1\right)^2}\times\frac{x\left(x^2+2x+1\right)}{x^3}\right)\div\frac{x-1}{x^3}\)
\(=\left(\frac{1}{\left(x+1\right)^2}\times\frac{x\left(x+1\right)^2}{x^3}\right)\div\frac{x-1}{x^3}\)
\(=\frac{1}{x^2}\times\frac{x^3}{x-1}\)
\(=\frac{x}{x-1}\)