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24 tháng 10 2019

<=> \(\left(x^3+3x^2a+3xa^2+a^3\right)-3bc\left(x+a\right)+b^3+c^3=0\)<=>\(\left(x+a\right)^3-3bc\left(x+a\right)+\left(b+c\right)^3-3bc\left(b+c\right)=0\)<=>

\(\left(x+a\right)^3+\left(b+c\right)^3-3bc\left(x+a+b+c\right)=0\)<=>

(x+a+b+c)[ (x+a)2 +(b+c)2 -(x+a)(b+c) -3bc]=0 <=> x+a+b+c=0 hoặc x2 + x(2a-b-c) + a2+ (b+c)2 -a(b+c)-3bc=0

<=> x= -a-b-c hoặc x2 + x(2a-b-c) + a2+ (b+c)2 -a(b+c)-3bc=0 (1)

\(\Delta=\left(2a-b-c\right)^2-4\left[a^2+\left(b+c\right)^2-a\left(b+c\right)-3bc\right]=\)\(4a^2+\left(b+c\right)^2-4a\left(b+c\right)-4a^2-4\left(b+c\right)^2+4a\left(b+c\right)\)\(+12bc=12bc-3\left(b+c\right)^2=-3\left(b-c\right)^2\le0\)

để (1) có nghiệm thì b-c=0 => \(\Delta=0\) => x = \(-\frac{2a-b-c}{2}=-a-b\)

kết luận

với b-c \(\ne0\) pt có 2 nghiệm x=-a-b-c

b-c=0 pt có 2 nghiệm x=-a-b-c và x=-a-b

24 tháng 2 2021

`a,(x+3)(x^2+2021)=0`

`x^2+2021>=2021>0`

`=>x+3=0`

`=>x=-3`

`2,x(x-3)+3(x-3)=0`

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

`=>x=+-3`

`b,x^2-9+(x+3)(3-2x)=0`

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

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

`=>` $\left[ \begin{array}{l}x=0\\x=-3\end{array} \right.$

`d,3x^2+3x=0`

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

`=>` $\left[ \begin{array}{l}x=0\\x=-1\end{array} \right.$

`e,x^2-4x+4=4`

`=>x^2-4x=0`

`=>x(x-4)=0`

`=>` $\left[ \begin{array}{l}x=0\\x=4\end{array} \right.$

1) a) \(\left(x+3\right).\left(x^2+2021\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x+3=0\\x^2+2021=0\end{matrix}\right.\\\left[{}\begin{matrix}x=-3\left(nhận\right)\\x^2=-2021\left(loại\right)\end{matrix}\right. \)

=> S={-3}

 

a: \(=\dfrac{\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)-3abc}{a^2+b^2+c^2-ab-bc-ac}\)

\(=\dfrac{\left(a+b+c\right)\left[\left(a+b\right)^2-c\left(a+b\right)+c^2\right]-3ab\left(a+b+c\right)}{a^2+b^2+c^2-ab-bc-ac}\)

\(=\dfrac{\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2-3ab\right)}{a^2+b^2+c^2-ab-bc-ac}\)

=a+b+c

b: 

Sửa đề: \(=\dfrac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)

\(=\dfrac{\left(x-y\right)^3+z^3+3xy\left(x-y\right)+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
\(=\dfrac{\left(x-y+z\right)\left(x^2-2xy+y^2-xz+yz+z^2\right)+3xy\left(x-y+z\right)}{2\left(x^2+y^2+z^2+xy+yz-xz\right)}\)

\(=\dfrac{\left(x-y+z\right)\left(x^2+y^2+z^2+xy-xz+yz\right)}{2\left(x^2+y^2+z^2+xy+yz-xz\right)}\)

\(=\dfrac{x-y+z}{2}\)

15 tháng 9 2023

a) \(\dfrac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ca}\)

\(=\dfrac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)}{a^2+b^2+c^2-ab-bc-ca}\)

\(=a+b+c\)

16 tháng 7 2017

b. Sử dụng các hằng đẳng thức

 \(a^3+b^3+c^2-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)\)

\(=3\left(a^2+b^2+c^2-ab-bc-ca\right)\)

và \(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3=3\left(a-b\right)\left(b-c\right)\left(c-a\right)\)

nên \(A=\frac{a^2+b^2+c^2-ab-bc-ca}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\frac{1}{2}.\frac{\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

Do (a - b) + (b - c) + (c - a) =  0 nên áp dụng hđt  \(X^2+Y^2+Z^2=-2\left(XY+YZ+ZX\right)\)khi X + Y + Z = 0, ta có:

\(A=-2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right).\)

16 tháng 7 2017

Bài 1 :

\(b,ax^2+3ax+9=a^2\) 

\(\Leftrightarrow a^2x+3ax+9-a^2=0\) 

\(\Leftrightarrow ax\left(a+3\right)+\left(a+3\right)\left(3-a\right)=0\) 

\(\Leftrightarrow\left(a+3\right)\left(ax+3-a\right)=0\)

Vì \(a\ne3\Rightarrow\left(a+3\right)\ne0\Rightarrow ax+3-a=0\) 

\(\Leftrightarrow ax=a-3\) 

Vì \(a\ne0\Rightarrow x=\frac{a-3}{a}\) 

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

b: \(\Leftrightarrow\left(x-3\right)\left(x+3\right)\left(x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-3\\x=4\end{matrix}\right.\)

c: \(\Leftrightarrow\left[{}\begin{matrix}2x+3=0\\5x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{3}{2}\\x=\dfrac{4}{5}\end{matrix}\right.\)

d: \(\Leftrightarrow\left(x+3\right)\left(x-4\right)=0\)

=>x+3=0 hoặc x-4=0

=>x=-3 hoặc x=4

e: \(\Leftrightarrow\left(x-3\right)\left(x+3\right)\left(x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-3\\x=4\end{matrix}\right.\)

f: \(\Leftrightarrow\left(2x+3\right)\left(x-4\right)\left(x+4\right)=0\)

hay \(x\in\left\{-\dfrac{3}{2};4;-4\right\}\)

8 tháng 2 2022

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

b, \(\Leftrightarrow\left[{}\begin{matrix}x^2-9=0\\4-x=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\pm3\\x=4\end{matrix}\right.\)

c, \(\Leftrightarrow\left[{}\begin{matrix}2x+3=0\\4-5x=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{3}{2}\\x=\dfrac{4}{5}\end{matrix}\right.\)

d, \(\Leftrightarrow\left[{}\begin{matrix}x+3=0\\x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\x=4\end{matrix}\right.\)

e, tương tự d 

f, \(\Leftrightarrow\left[{}\begin{matrix}2x+3=0\\x^2-16=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{3}{2}\\x=\pm4\end{matrix}\right.\)