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b ) x2 - 4x - 2y + xy + 1 = 0
( x2 - 4x + 4 ) - y ( 2 - x ) -3 = 0
( x - 2 )2 - y ( 2 - x ) = 3
( 2 - x ) ( 2 - x - y ) = 3
đến đây lập bảng tìm ra x,y
a) x2 + y2 + xy + 3x - 3y + 9 = 0
2x2 + 2y2 + 2xy + 6x - 6y + 18 = 0
( x2 + 2xy + y2 ) + ( x2 + 6x + 9 ) + ( y2 - 6y + 9 ) = 0
( x + y )2 + ( x + 3 )2 + ( y - 3 )2 = 0
\(\Rightarrow\)( x + y )2 = ( x + 3 )2 = ( y - 3 )2 = 0
\(\Rightarrow\)x = -3 ; y = 3
Câu 4:
5x + 7y = 112
5(x+ y) = 112 - 2y
5(x + y) = 2(56 - y)
\(\begin{cases}x+y=2\\ 56-y=5\end{cases}\)
\(\begin{cases}x+y=2\\ y=56-5\end{cases}\)
\(\begin{cases}x=2-y\\ y=51\end{cases}\)
\(\begin{cases}x=2-51\\ y=51\end{cases}\)
\(\begin{cases}x=-49\\ y=51\end{cases}\)
Vậy (x ; y) = (-49; 51)
1)<=> \(\left(x^2+2x+1\right)-\left(y^2+4x+4\right)-7=0\)
=> \(\left(x+1\right)^2-\left(y+2\right)^2=7\)
\(\Rightarrow\left(x+1-y-2\right)\left(x+1+y+2\right)=7\)
\(\left(x-y-1\right)\left(x+y+3\right)=7\)
vì x;y là số nguyên dương=> \(x;y\ge1\Rightarrow x+y+3\ge1+1+3=5\)
=> \(x+y+3=7\Rightarrow x+y=4\)
\(x-y-1=1\Rightarrow x-y=2\)
cộng hai phương trình ta có:
=> \(\left(x+y\right)+\left(x-y\right)=4+2\)
=> \(2x=6\)
=> x=3
=> y=4-3=1
2)<=> \(\left(x^2+xy+2x\right)+\left(2xy+2y^2+4y\right)+\left(x+y+2\right)=17\)
=> \(x\left(x+y+2\right)+2y\left(x+y+2\right)+1\left(x+y+2\right)=17\)
=> \(\left(x+y+2\right)\left(x+2y+1\right)=17\)
x+y+2 | x+2y+1 | hệ phương trình | nghiệm (x;y) |
1 | 17 | x+y=-1 và x+2y=16 | (-18;17) |
17 | 1 | x+y=15 và x+2y=0 | (30;-15) |
-1 | -17 | x+y=-3 và x+2y=-18 | (12;-15) |
-17 | -1 | x+y=-19 và x+2y=-2 | (-36;17) |
\(a,x^2+y^2-x-y=8\)
\(\Rightarrow x^2-x+\frac{1}{4}+y^2-y+\frac{1}{4}-8,5=0\)
\(\Rightarrow\left(x-\frac{1}{2}\right)^2+\left(y-\frac{1}{2}\right)^2-8,5=0\)
Ta có : \(\left(x-\frac{1}{2}\right)^2+\left(y-\frac{1}{2}\right)^2-8,5\ge-8,5\forall x;y\)
Để VP=0 và là các số nguyên
=>\(\left(x-\frac{1}{2}\right)^2+\left(y-\frac{1}{2}\right)^2=8,5\)
a/ x^2 + y^2 - x - y = 8
<=> 4x^2 + 4y^2 - 4x - 4y = 32
<=> (2x - 1)^2 + (2y - 1)^2 = 34
<=> (2x - 1)^2 = 9 và (2y - 1)^2 = 25
Hoặc (2x - 1)^2 = 25 và (2y - 1)^2 = 9
Câu 1:
\(\text{a) }\dfrac{x^2-xy}{3xy-3y^2}=\dfrac{x\left(x-y\right)}{3y\left(x-y\right)}=\dfrac{x}{3y}\)
\(\text{b) }\dfrac{2ax^2-4ax+2a}{5b-5bx^2}\\ =\dfrac{2a\left(x^2-2x+1\right)}{5b\left(1-x^2\right)}\\ =\dfrac{2a\left(x-1\right)^2}{5b\left(1-x\right)\left(1+x\right)}\\ =-\dfrac{2a\left(x-1\right)^2}{5b\left(x-1\right)\left(1+x\right)}\\ =-\dfrac{2a\left(x-1\right)}{5b\left(x+1\right)}\\ =-\dfrac{2ax-2a}{5bx+5b}\)
\(\text{c) }\dfrac{4x^2-4xy}{5x^3-5x^2y}=\dfrac{4x\left(x-y\right)}{5x^2\left(x-y\right)}=\dfrac{4}{5x}\)
\(\text{d) }\dfrac{\left(x+y\right)^2-z^2}{x+y+z}=\dfrac{\left(x+y+z\right)\left(x+y-z\right)}{x+y+z}=x+y-z\)
\(\text{e) }\dfrac{x^6+2x^3y^3+y^6}{x^7-xy^6}\\ =\dfrac{\left(x^3+y^3\right)^2}{x\left(x^6-y^6\right)}\\ =\dfrac{\left(x^3+y^3\right)^2}{x\left(x^3-y^3\right)\left(x+y\right)^3}\\ =\dfrac{x^3+y^3}{x\left(x^3-y^3\right)}\\ =\dfrac{x^3+y^3}{x^4-xy^3}\)
Câu 3:
\(\text{ a) }\dfrac{\left(a+b\right)^2-c^2}{a+b+c}=\dfrac{\left(a+b+c\right)\left(a+b-c\right)}{a+b+c}=a+b-c\)
\(\text{b) }\dfrac{a^2+b^2-c^2+2ab}{a^2-b^2+c^2+2ac}\\ =\dfrac{\left(a^2+2ab+b^2\right)-c^2}{\left(a^2+2ac+c^2\right)-b^2}\\ =\dfrac{\left(a+b\right)^2-c^2}{\left(a+c\right)^2-b^2}\\ =\dfrac{\left(a+b+c\right)\left(a+b-c\right)}{\left(a+c+b\right)\left(a+c-b\right)}\\ =\dfrac{a+b-c}{a-b+c}\)
\(\text{c) }\dfrac{2x^3-7x^2-12x+45}{3x^3-19x^2+33x-9}\\ =\dfrac{2x^3-x^2-6x^2+3x-15x+45}{3x^3-10x^2-9x^2+3x+30x-9}\\ =\dfrac{\left(2x^3-x^2-15x\right)-\left(6x^2-3x-45\right)}{\left(3x^3-10x^2+3x\right)-\left(9x^2-30x+9\right)}\\ =\dfrac{x\left(2x^2-x-15\right)-3\left(2x^2-x-15\right)}{x\left(3x^2-10x+3\right)-3\left(3x^2-10x+3\right)}\\ =\dfrac{\left(x-3\right)\left(2x^2-x-15\right)}{\left(x-3\right)\left(3x^2-10x+3\right)}\\ =\dfrac{\left(x-3\right)\left(2x^2-6x+5x-15\right)}{\left(x-3\right)\left(3x^2-9x-x+3\right)}\\ =\dfrac{\left(x-3\right)\left[\left(2x^2-6x\right)+\left(5x-15\right)\right]}{\left(x-3\right)\left[\left(3x^2-9x\right)-\left(x-3\right)\right]}\\ =\dfrac{\left(x-3\right)\left[x\left(x-3\right)+5\left(x-3\right)\right]}{\left(x-3\right)\left[3x\left(x-3\right)-\left(x-3\right)\right]}\\ =\dfrac{\left(x-3\right)\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x-3\right)\left(3x-1\right)}\\ =\dfrac{x+5}{3x-1}\)