Bài 1:
a) \(x^2-10x+26+y^2+2y\)
\(=x^2-2.x.5+25+y^2+2y+1\)
\(=\left(x-5\right)^2+\left(y+1\right)^2\)
b) Sửa đề \(z^2-6z+5-t^2-4t\)
\(=z^2-2.z.3+9-4-t^2-4t\)
\(=\left(z-3\right)^2-\left(t^2+4t+4\right)\)
\(=\left(z-3\right)^2-\left(t+2\right)^2\)
c) \(\left(x+y-4\right)\left(x+y+4\right)\)
\(=\left(x+y\right)^2-4^2\)
d) \(a^2-b^2+c^2-2ac-d^2+2bd\)
\(=\left(a^2-2ac+c^2\right)-\left(b^2-2bd+d^2\right)\)
\(=\left(a-c\right)^2-\left(b-d\right)^2\)
e) \(\left(a-b-c\right)\left(a+b-c\right)\)
\(=\left(a-c-b\right)\left(a-c+b\right)\)
\(=\left(a-c\right)^2-b^2\)
f) \(4a^2+2b^2-4ab-2b+1\)
\(=\left(2a\right)^2-2.2a.b+b^2+b^2-2b+1\)
\(=\left(2a-b\right)^2+\left(b-1\right)^2\)
Bài 2:
a) Sửa đề \(4x^2-4xy+y^2\)
\(=\left(2x\right)^2-2.2x.y+y^2\)
\(=\left(2x-y\right)^2\)
b) \(y^2-6y+9\)
\(=y^2-2.y.3+3^2\)
\(=\left(y-3\right)^2\)
c) \(a^2+a+\dfrac{1}{4}\)
\(=a^2+2a.\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2\)
\(=\left(a+\dfrac{1}{2}\right)^2\)
d) \(a^2-12a+36\)
\(=a^2-2.a.6+6^2\)
\(=\left(a-6\right)^2\)
i) \(x^2-xy+\dfrac{1}{4}y^2\)
\(=x^2-2.x.\dfrac{1}{2}y+\left(\dfrac{1}{2}y\right)^2\)
\(=\left(x-\dfrac{1}{2}y\right)^2\)
e) \(9x^2-24x+16\)
\(=\left(3x\right)^2-2.3x.4+4^2\)
\(=\left(3x-4\right)^2\)
f) \(x^2-3x+\dfrac{9}{4}\)
\(=x^2-2.x.\dfrac{3}{2}+\left(\dfrac{3}{2}\right)^2\)
\(=\left(x-\dfrac{3}{2}\right)^2\)
g) \(1-2xy^2+x^2y^4\)
\(=1-2xy^2+\left(xy^2\right)^2\)
\(=\left(1-xy^2\right)^2\)
h) \(\left(2a-b\right)^2+2\left(2a-b\right)+1\)
\(=\left(2a-b+1\right)^2\)
Bài 3:
a) \(A=\dfrac{1}{4}x^2-xy+y^2\)
\(A=\left(\dfrac{1}{2}x\right)^2-2.\dfrac{1}{2}x.y+y^2\)
\(A=\left(\dfrac{1}{2}x-y\right)^2\)
Thay x = 2012 và y = 1004 vào A ta được
\(A=\left(\dfrac{1}{2}.2012-1004\right)^2\)
\(A=\left(1006-1004\right)^2\)
\(A=2^2=4\)
b) \(B=9x^2-3xy+\dfrac{1}{4}y^2\)
\(B=\left(3x\right)^2-2.3x.\dfrac{1}{2}y+\left(\dfrac{1}{2}y\right)^2\)
\(B=\left(3x-\dfrac{1}{2}y\right)^2\)
Thay x = 231 và y = 1384 vào B ta được
\(B=\left(3.231-\dfrac{1}{2}.1384\right)^2\)
\(B=\left(693-692\right)^2\)
\(B=1^2=1\)
