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a)x2+2x+1=x2+2x.1+12=(x+1)2
b)x2-x+\(\frac{1}{4}\)=x2-2.x.\(\frac{1}{2}\)+\(\left(\frac{1}{2}\right)^2\)=\(\left(x-\frac{1}{2}\right)^2\)
1) viết biểu thức dưới dạng bình phương của 1 tổng hoặc 1 hiệu
a) x2 - 2x +1
= x2 - 2 . x . 1 + 12
= ( x + 1 )2
b) 81x2 + 1 +18x
= 81x2 + 18x + 1
= (9x)2 + 2 . 9x . 1 + 12
= (9x + 1)2
c)1/4 + x2 + x
= x2 + 2 . x . 1/2 + (1/2)2
= (x + 1/2)2
~.~
1a/ z2 - 6z + 5 - t2 - 4t = z2 - 2 . 3z + 32 - 4 - t2 - 4t = (z2 - 2 . 3z + 32) - (22 + 2 . 2t + t2) = (z - 3)2 - (2 + t)2
b/ x2 - 2xy + 2y2 + 2y2 + 1 = x2 - 2xy + y2 + y2 + 2y + 1 = (x2 - 2xy + y2) + (y2 + 2y + 1) = (x - y)2 + (y + 1)2
c/ 4x2 - 12x - y2 + 2y + 8 = (2x)2 - 12x - y2 + 2y + 32 - 1 = [ (2x)2 - 2 . 3 . 2x + 32 ] - (y2 - 2y + 1) = (2x - 3)2 - (y - 1)2
2a/ (x + y + 4)(x + y - 4) = x2 + xy - 4x + xy + y2 - 4y + 4x + 4y + 16 = x2 + (xy + xy) + (-4x + 4x) + (-4y + 4y) + y2 + 16
= x2 + 2xy + y2 + 42 = (x + y)2 + 42
b/ (x - y + 6)(x + y - 6) = x2 + xy - 6x - xy - y2 + 6y + 6x + 6y - 36 = x2 + (xy - xy) + (-6x + 6x) + (6y + 6y) - y2 - 36
= x2 - y2 + 12y - 62 = x2 - (y2 - 12y + 62) = x2 - (y2 - 2 . 6y + 62) = x2 - (y - 6)2
c/ (y + 2z - 3)(y - 2z - 3) = y2 -2yz - 3y + 2yz - 4z2 - 6z - 3y + 6z + 9 = y2 + (-2yz + 2yz) + (-3y - 3y) + (-6z + 6z) - 4z2 + 9
= y2 - 6y - 4z2 + 9 = (y2 - 6y + 9) - 4z2 = (y - 3)2 - (2z)2
d/ (x + 2y + 3z)(2y + 3z - x) = 2xy + 3xz - x2 + 4y2 + 6yz - 2xy + 6yz + 9z2 - 3xz = (2xy - 2xy) + (3xz - 3xz) - x2 + (6yz + 6yz) + 9z2 + 4y2
= -x2 + 4y2 + 12yz + 9z2 = (4y2 + 12yz + 9z2) - x2 = [ (2y)2 + 2 . 2 . 3yz + (3z)2 ] - x2 = (2y + 3z)2 - x2
3.
a, (2y- 1)3= (2y)3-3.(2y)2.1+3.2y.12-13
= 8y3-12y2+6y-1
b, (3x2+2y)3=(3x2)3+3.(3x2)2.2y+3.3x2.(2y)2+13
=27x6+54x4y+36x2y2+1
c, ( 1/3x-2)3=(1/3x)3-3.(1/3x)2.2+3.1/3x.22-23
=1/27x3-2/3x2+4x-8
4.
a, -x3+3x3-3x+1=1-3x+3x3-x3
=1-3.12.x+3.1.x3-x3
=(1-x)3
b,64-48x+12x2-x3=43-3.42.x+3.4.x2-x3
=(4-x)3
Bài 3 Tính:
\(a\)) \(\left(2y-1\right)^3=2y^3-3.\left(2y\right)^2.1+3.2y.1^2-1^3\)
\(=2y^3-12y^2+6y-1\)
b)\(\left(3x^2+2y\right)^3\)
\(=\left(3x^2\right)^3=3.\left(3x^2\right)^2.2y+3.\left(3x^2\right).\left(2y\right)^2+\left(2y\right)^3\)
\(=27x^8+3.9x^4.2+9x^2.4y+8y^3\)
\(=27x^8+54x^4+36x^2y+8y^3\)
c)\(\left(\dfrac{1}{3}x-2\right)^3\)
\(=\left(\dfrac{1}{3}x\right)^3-3.\left(\dfrac{1}{3}x\right)^2.2+3.\dfrac{1}{3}x.2^2-2^3\)
\(=\dfrac{1}{27}x^3-3.\dfrac{1}{9}x^2.2+x.2^2-8\)
\(=\dfrac{1}{27}x^3-\dfrac{2}{3}x^2+4x-8\)
a, \(25x^2+5xy+\frac{1}{4}y^2=\left(5x\right)^2+2.5x.\frac{1}{2}y+\left(\frac{1}{2}y\right)^2\)
\(=\left(5x+\frac{1}{2}y\right)^2\)
b, \(9x^2+12x+4=\left(3x\right)^2+2.3x.2+2^2=\left(3x+2\right)^2\)
c, \(x^2-6x+5-y^2-4y=\left(x^2-6x+9\right)-\left(y^2+4y+4\right)\)
\(=\left(x-3\right)^2-\left(y+2\right)^2=\left(x-y-5\right)\left(x+y-1\right)\)
d, \(\left(2x-y\right)^2+4\left(x+y\right)^2-4\left(2x-y\right)\left(x+y\right)\)
\(=\left(2x-y\right)^2-2\left(2x-y\right)\left(2x+2y\right)+\left(2x+2y\right)^2\)
\(=\left(2x-y+2x+2y\right)^2=\left(4x+y\right)^2\)