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a) x2+y2-4x+4y+8=0
⇔ (x-2)2+(y+2)2=0
\(\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\y+2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=-2\end{matrix}\right.\)
b)5x2-4xy+y2=0
⇔ x2+(2x-y)2=0
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\2x-y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)
c)x2+2y2+z2-2xy-2y-4z+5=0
⇔ (x-y)2+(y-1)2+(z-2)2=0
\(\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\y-1=0\\z-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y=1\\z=2\end{matrix}\right.\)
b: Ta có: \(5x^2-4xy+y^2=0\)
\(\Leftrightarrow x^2-\dfrac{4}{5}xy+y^2=0\)
\(\Leftrightarrow x^2-2\cdot x\cdot\dfrac{2}{5}y+\dfrac{4}{25}y^2+\dfrac{21}{25}y^2=0\)
\(\Leftrightarrow\left(x-\dfrac{2}{5}y\right)^2+\dfrac{21}{25}y^2=0\)
Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)
1/x^3 - 2x^2 - 9x + 18
= x\(^2\)( x - 2 ) - 9 ( x - 2 ) = ( x\(^2\) - 9 ) ( x - 2 )= ( x - 3 ) ( x +3 ) ( x - 2 )
2/3x^2 -5x - 3y^2 + 5y
= 3( x\(^2\) - y\(^2\) ) - 5 ( x - y ) = 3 ( x - y ) ( x + y ) - 5 ( x - y ) = ( x - y ) [ 3( x+ y ) - 5 ]
= ( x - y ) ( 3x + 3y - 5 )
3/49 - x^2 + 2xy - y^2
= 49 - ( x\(^2\) - 2xy + y\(^2\) ) = 49 - ( x - y )\(^2\) = ( 7 - x + y ) ( 7 + x - y )
5/ x^2 - 4x^2y^2 + 2xy
= x ( x - 4xy\(^2\) + 2y )
6/ 3x - 3y - x^2 + 2xy - y^2
= ( 3x - 3y ) - ( x\(^2\) - 2xy + y\(^2\) ) = 3 ( x - y ) - ( x - y )\(^2\) = ( x - y ) ( 3 - x + y )
1)2xy+3z+6y+xz
= x(2y + z) + 3(z + 2y)
= (x + 3)(2y + z)
2)x^4-9x^3+x^2-9x
= x^2(x^2 + 1) - 9x(x^2 + 1)
= (x^2 + 1)(x^2 - 9x)
= x(x - 9)(x^2 + 1)
3)x^2-xy+x-y
= x(x - y) + (x - y)
= (x + 1)(x - y)
4)xz+yz-5(x+y)
= z(x + y) - 5(x + y)
= (z - 5)(x + y)
5)3x^2-3xy-5x+5y
= 3x(x - y) - 5(x - y)
= (3x - 5)(x - y)
6)x^2+4x-y^2+4y
= (x - y)(x + y) + 4(x + y)
= (x - y + 4)(x + y)
1) x2 + x - y2 + y = (x2 - y2) + (x + y) = (x - y)(x + y) + (x + y) = (x - y + 1)(x + y)
2) 4x2 - 9y2 + 4x - 6y = (4x2 - 9y2) + (4x - 6y) = (2x - 3y)(2x + 3y) + 2(2x - 3y) = (2x - 3y)(2x + 3y + 2)
3) x2 + x + y2 + y + 2xy = (x2 + 2xy + y2) + (x + y) = (x + y)2 + (x + y) = (x + y)(x + y + 1)
4) -x2 + 5x + 2xy - 5y - y2 = -(x2 - 2xy + y2) + (5x - 5y) = -(x - y)2 + 5(x - y) = (x - y)(y - x + 5)
5) x2 - y2 + 2x + 1 = (x2 + 2x + 1) - y2 = (x + 1)2 - y2 = (x + 1 + y)(x - y + 1)
6) x2 - 1 - y2 + 2y = x2 - (y2 - 2y + 1) = x2 - (y - 1)2 = (x - y + 1)(x + y - 1)
7) x2 + 2xz - y2 + 2uy + z2 - u2 =(x2 + 2xz + z2) - (y2 - 2uy + u2) = (x + z)2 - (y - u)2 = (x + z - y + u)(x + z + y - u)
8) x3 + 3x2y + x + 3xy2 + y + y3 = (x3 + 3x2y + 3xy2 + y3) + (x + y) = (x + y)3 + (x + y) = (x + y)(x2 + 2xy + y2 + 1)
9) x3 + y(1 - 3x2) + x(3y2 - 1) - y3 = x3 + y - 3x2y + 3xy2 - x - y3 = (x3 - 3x2y + 3xy2 - y3) - (x - y) = (x - y)3 - (x - y) = (x - y)(x2 - 2xy+y2-1)
1) \(5x-5y+x\left(x-y\right)\)
\(=5\left(x-y\right)+x\left(x-y\right)\)
\(=\left(x-y\right)\left(x+5\right)\)
2) \(x^2+4x+3\)
\(=\left(x^2+x\right)+\left(3x+3\right)\)
\(=x\left(x+1\right)+3\left(x+1\right)\)
\(=\left(x+1\right)\left(x+3\right)\)
3) \(x^2-2xy+y^2-z^2\)
\(=\left(x-y\right)^2-z^2\)
\(=\left(x-y-z\right)\left(x-y+z\right)\)
4) \(x\left(x-5\right)-3x+15\)
\(=x\left(x-5\right)-3\left(x-5\right)\)
\(=\left(x-5\right)\left(x-3\right)\)
5) \(y^2-x^2+2x-1\)
\(=y^2-\left(x^2-2x+1\right)\)
\(=y^2-\left(x-1\right)^2\)
\(=\left(x+y-1\right)\left(y-x+1\right)\)
\(1.\left(x-y\right)\left(x+5\right)\)
\(2.\left(x+1\right)\left(x+3\right)\)
\(3.\left(x-y-z\right)\left(x-y+z\right)\)
\(4.\left(x-3\right)\left(x-5\right)\)
\(5.\left(y-x+1\right)\left(y+x+1\right)\)
\(7.\left(x+1\right)\left(x-2\right)^2\)
\(8.\left(x-5\right)\left(x+3\right)\)
\(10.\left(y+1\right)\left(2x+z\right)\)
1)
5x - 5y + x ( x - y ) = (x-y)(5+x)
2)
x2+4x+3=x2+x+3x+3=(x+1)(x+3)
3)x2-2xy+y2-z2=\(\left(x-y\right)^2-z^2=\left(x-y-z\right)\left(x-y+z\right)\)
4)\(x\left(x-5\right)-3x+15=\left(x-3\right)\left(x-5\right)\)
\(\text{a) }\left(x-1\right)\left(x^2+y\right)-\left(x^2-y\right)\left(x-2\right)-x\left(x+2y\right)+3\left(y-5\right)\)
\(=\left(x^3+xy-x^2-y\right)-\left(x^3-2x^2-xy+2y\right)-\left(x^2+2xy\right)+\left(3y-15\right)\)
\(=x^3+xy-x^2-y-x^3+2x^2+xy-2y-x^2-2xy+3y-15\)
\(=\left(x^3+x^3\right)+\left(-x^2+2x^2-x^2\right)+\left(xy+xy-2xy\right)+\left(-y-2y+3y\right)-15\)
\(=0+0+0+0-15\)
\(=-15\)
\(\text{b) }6\left(x^3y+x-3\right)-6x\left(2xy^3+1\right)-3x^2y\left(2x-4y^2\right)\)
\(=\left(6x^3y+6x-18\right)-\left(12x^2y^3+6x\right)-\left(6x^3y-12x^2y^3\right)\)
\(=6x^3y+6x-18-12x^2y^3-6x-6x^3y+12x^2y^3\)
\(=\left(6x^3y-6x^3y\right)+\left(6x-6x\right)+\left(-12x^2y^3+12x^2y^3\right)-18\)
\(=0+0+0-18\)
\(=-18\)
\(\text{c) }\left(x^2+2xy+4y^2\right)\left(x-2y\right)-6\left(\frac{1}{2}-\frac{4}{3}y^3\right)\)
\(=\left(x^3-2x^2y+2x^2y-4xy^2+4xy^2-8y^3\right)-\left(3-8y^3\right)\)
\(=\left(x^3-8y^3\right)-\left(3-8y^3\right)\)
\(=x^3-8y^3-3+8y^3\)
\(=x^3-3\)