Cho a+b+c=2x.Cm:2ab+b^2+c^2-a^2 = 4x(x-a)
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29 tháng 1 2022
Bài 1:
a: \(4x^2-4x-2=4x^2-4x+1-3=\left(2x-1\right)^2-3>=-3\forall x\)
Dấu '=' xảy ra khi x=1/2
b: \(x^4+4x^2+1>=1\forall x\)
Dấu '=' xảy ra khi x=0
c: \(2x^2-20x-7\)
\(=2\left(x^2-10x-\dfrac{7}{2}\right)\)
\(=2\left(x^2-10x+25-\dfrac{57}{2}\right)\)
\(=2\left(x-5\right)^2-57>=-57\forall x\)
Dấu '=' xảy ra khi x=5
6 tháng 8 2023
15:
a: (a-b)^2
=a^2-2ab+b^2
=a^2+2ab+b^2-4ab
=(a+b)^2-4ab
b: (x+y)^2+(x-y)^2
=x^2+2xy+y^2+x^2-2xy+y^2
=2x^2+2y^2
=2(x^2+y^2)
14:
P=9x^2-12x+4
=(3x)^2-2*3x*2+2^2
=(3x-2)^2
a: Khi x=14 thì P=(3*14-2)^2=40^2=1600
b: Khi x=2/3 thì P=(3*2/3-2)^2=0
c: Khi x=-8/3 thì P=(-8/3*3-2)^2=(-10)^2=100
12:
a: 101^2=(100+1)^2
=100^2+2*100+1
=10000+200+1
=10201
b: 75^2-50*75+25^2
=(75-25)^2=50^2=2500
c: 103*97
=(100+3)(100-3)
=100^2-9
=9991
6 tháng 8 2023
Bài 1: b) B = (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc Bài 2: a) (x - 1/3)^2 = x^2 - 2/3x + 1/9 b) (x + y^2/3)^3 = x^3 + 3x^2y^2/3 + 3xy^4/9 + y^6/27 Bài 3: a) (2x + 1)(4x^2 - 2x + 1) = 8x^3 + 4x^2 - 2x + 4x^2 - 2x + 1 = 8x^3 + 8x^2 - 4x + 1 b) (1 - x^2)(1 + x^2 + x^4) = 1 - x^4 + x^2 + x^2 - x^4 + x^6 = x^6 - 2x^4 + 2x^2 + 1 c) (y - x/y)(y^2 + x + x^2/y^2) = y^3 + xy - x^2 + y^2 + xy + x^2 + x/y^2 - x - x^2/y^2 = y^3 + 2xy + y^2 - x Bài 4: a) M = (x + 3)(x^2 - 3x + 9) = x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 = x^3 + 27 b) N = (1 - 3x)(1 + 3x + 9^2) = 1 - 9x + 3x - 27x^2 + 9 + 27x + 81 = -27x^2 + 27 c) P = (x - 1/2)(x^2 + x/2 + 1/4) = x^3 + x^2/2 + x^2/4 - x/2 - x/4 + 1/8 = x^3 + 3x^2/4 - 3x/4 + 1/8 d) Q = (2x + 3y)(4x^2 - 6xy + 9y^2) = 8x^3 - 12x^2y + 18xy^2 + 12x^2y - 18xy^2 + 27y^3 = 8x^3 + 27y^3 Bài 5: a) x^2 + 6x + 9 = (x + 3)^2 b) 9x^2 - 6x + 1 = (3x - 1)^2 c) x^2y^2 + xy + 1/4 = (xy + 1/2)^2 d) (x - y)^2 + 6(x - y) + 9 = (x - y + 3)^2
Bài 6: a) x^2 + 6x + 9 = (x + 3)^2 b) 4x^2 - 4x + 1 = (2x - 1)^2 c) 9x^2 - 12xy + 4y^2 = (3x - 2y)^2 d) (x - y)(x + y/3) = x^2 - y^2/9 Bài 7: a) -x^3 + 3x^2 - 3x + 1 = -(x - 1)^3 b) x^3 + x^2 + 1/3x + 1/27 = (x + 1/3)^3 c) x^6 - 3^4y + 3^2y^2 - y^3 = (x^2 - 3y)^3 d) (x - y)^3 + (x - y)^2 + 1/3(x - y) + 1/27 = (x - y + 1/3)^3 Bài 8: a) x^3 + 27 = (x + 3)(x^2 - 3x + 9) b) x^3 - 1/8 = (x - 1/2)(x^2 + 1/2x + 1/4) c) 8x^3 + y^3 = (2x)^3 + y^3 = (2x + y)(4x^2 - 2xy + y^2) d) 8^3 - 27y^3 = 512 - 27y^3 = (8 - 3y)(64 + 24y + 9y^2) Bài 9: a) A = -(-28)^3 + 6(-28)^2 - 12(-28) + 8 = -21952 b) B = 8(1/2)^3 + 12(1/2)^2 + 6(1/2) + 1 = 2 c) C = (20 + 2(1))^3 - 6(20 + 2(1))^2 + 12(20 + 2(1)) - 8 = 0 Bài 10: a) 11^3 - 1 = 1330 b) x^3 - y^3 = (x - y)(x^2 + xy + y^2) = 6(6^2 + 9) = 450Bài 11: a) M = (x + 3)(x^2 - 3x + 9) - (3 - 2x)(4x^2 + 6x + 9) = x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 - (12x^2 + 18x + 27 - 8x^2 - 12x - 18) = x^3 - 12x^2 + 9x + 27 - 4x^2 - 12x + 9 = x^3 - 16x^2 - 3x + 36 b) N = (x - 2y)(x^2 + 2xy + 4y^2) + 16y^3 = (x - 2y)(x^2 + 2xy + 4y^2) + 16y^3 = x^3 - 2x^2y + 2xy^2 - 4y^3 + 16y^3 = x^3 - 2x^2y + 2xy^2 + 12y^3 Bài 12: a) 101^2 = 10201 b) 75^2 - 50.75 + 25^2 = 5625 - 3750 + 625 = 2500 c) 103.97 = 10091 Bài 13: a) 101^3 = 1030301 b) 98^3 + 6.98^2 + 12.98 + 8 = 941192 c) 99^3 = 970299 Bài 14: a) P = 9(34)^2 - 12(34) + 4 = 8296 b) P = 9(2/3)^2 - 12(2/3) + 4 = 0 c) P = 9(-8/3)^2 - 12(-8/3) + 4 = 128 Bài 15: a) (a - b)^2 = (a + b)^2 - 4ab = a^2 - 2ab + b^2 = a^2 + 2ab + b^2 - 4ab = (a + b)^2 - 4ab b) (x + y)^2 + (x - y)^2 = 2(x^2 + y^2) = x^2 + 2xy + y^2 + x^2 - 2xy + y^2 = 2x^2 + 2y^2
ND
Nguyễn Đức Trí
VIP
25 tháng 8 2023
a) \(x^2+2x+1=\left(x+1\right)^2\)
b) \(x^2+8x+16=\left(x+4\right)^2\)
c) \(x^2+6x+9=\left(x+3\right)^2\)
d) \(4x^2+4x+1=\left(2x+1\right)^2\)
e) \(36+x^2-12x=x^2-12x+36=\left(x-6\right)^2\)
f) \(4x^2+12x+9=\left(2x+3\right)^2\)
g) \(x^4+81+18x^2=x^4+18x^2+81=\left(x^2+9\right)^2\)
h) \(9x^2+30xy+25y^2=\left(3x+5y\right)^2\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
25 tháng 8 2023
a, \(x^2\) + 2\(x\) + 1 = (\(x\) + 1)2
b, \(x^2\) + 8\(x\) + 16 = (\(x\) + 4)2
c, \(x^2\) + 6\(x\) + 9 = (\(x\) + 3)2
d, 4\(x^2\) + 4\(x\) + 1 = (2\(x\) + 1)2
14 tháng 8 2023
a: =x^2+2xy+y^2-4x^2y^2
=(x+y)^2-(2xy)^2
=(x+y+2xy)(x+y-2xy)
b: =49-(a^2-2ab+b^2)
=49-(a-b)^2
=(7-a+b)(7+a-b)
c: =\(a^2-\left(b^2-4bc+4c^2\right)\)
\(=a^2-\left(b-2c\right)^2=\left(a-b+2c\right)\left(a+b-2c\right)\)
d:
\(=\left(bc\right)^2-\left(b^2+c^2-a^2\right)^2\)
\(=\left(bc-b^2-c^2+a^2\right)\left(bc+b^2+c^2-a^2\right)\)
e: \(=\left(a+b\right)^2+2c\left(a+b\right)+c^2+\left(a+b\right)^2-2c\left(a+b\right)+c^2-4c^2\)
=2(a+b)^2-2c^2
=2[(a+b)^2-c^2]
=2(a+b-c)(a+b+c)
25 tháng 8 2023
`#040911`
`a)`
`196 - a^2 + 2ab - b^2`
`= 196 - (a^2 - 2ab + b^2)`
`= 196 - (a - b)^2`
`= 14^2 - (a - b)^2`
`= (14 - a + b)(14 + a - b)`
`b)`
`a^2 + 6a - 4b^2 + 9`
`= (a^2 + 6a + 9) - 4b^2`
`= [ (a)^2 + 2*a*3 + 3^2] - (2b)^2`
`= (a + 3)^2 - (2b)^2`
`= (a + 3 - 2b)(a + 3 + 2b)`
`c)`
`4x - 4 + 9y^2 - x^2`
`= 9y^2 - (x^2 - 4x + 4)`
`= (3y)^2 - [ (x)^2 - 2*x*2 + 2^2]`
`= (3y)^2 - (x - 2)^2`
`= (3y - x + 2)(3y + x - 2)`
`d)`
`5x^2 - 10x + 5 - 45t^2`
`= 5*(x^2 - 2x + 1 - 9t^2)`
`= 5*[ (x^2 - 2x + 1) - 9t^2]`
`= 5*{ [(x)^2 - 2*x*1 + 1^2] - (3t)^2}`
`= 5*[ (x - 1)^2 - (3t)^2]`
`= 5*(x - 1 - 3t)(x - 1 + 3t)`
`e)`
`x^2 - 36y^2t^2 - 10x +25`
`= (x^2 - 10x + 25) - 36y^2t^2`
`= [ (x)^2 - 2*x*5 + 5^2] - (6yt)^2`
`= (x - 5)^2 - (6yt)^2`
`= (x - 5 - 6yt)(x - 5 + 6yt)`
25 tháng 8 2023
a: =196-(a^2-2ab+b^2)
=196-(a-b)^2
=(14-a+b)(14+a-b)
b: \(=\left(a^2+6a+9\right)-4b^2\)
\(=\left(a+3\right)^2-4b^2\)
\(=\left(a+3-2b\right)\left(a+3+2b\right)\)
c: \(=9y^2-\left(x^2-4x+4\right)\)
\(=\left(3y\right)^2-\left(x-2\right)^2\)
\(=\left(3y-x+2\right)\left(3y+x-2\right)\)
d: \(=5\left(x^2-2x+1-9t^2\right)\)
\(=5\left[\left(x-1\right)^2-\left(3t\right)^2\right]\)
\(=5\left(x-1-3t\right)\cdot\left(x-1+3t\right)\)
e: \(=x^2-10x+25-36y^2t^2\)
\(=\left(x-5\right)^2-\left(6yt\right)^2\)
\(=\left(x-5-6yt\right)\left(x-5+6yt\right)\)
S
18 tháng 10 2018
a, \(5a^2-5b^2=5\left(a^2-b^2\right)=5\left(a+b\right)\left(a-b\right)\)
b, \(2ab^2-a^2b-b^3=b\left(2ab-a^2b-b^2\right)=-b\left(a^2-2ab+b^2\right)\)
\(=-b\left(a-b\right)^2\)
18 tháng 10 2018
a) 5a2 - 5b2 = 5(a2-b2) = ( a+b) (a-b)
b) 2ab2 - a2b - b3 = b ( 2ab-a2-b2 ) = -b (a2-2ab+b2) = -b (a-b)2
Ta có: \(a+b+c=2x\)
\(\Rightarrow\left(a+b+c\right)^2=4x^2\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=4x^2\)
\(VP=4x\left(x-a\right)\)
\(=4x^2-4xa\)
\(=a^2+b^2+c^2+2ab+2bc+2ac-\left(2a+2b+2c\right)a\)
\(=a^2+b^2+c^2+2ab+2bc+2ac-2a^2-2ab-2ac\)
\(=2bc+b^2+c^2-a^2=VT\)
\(\Rightarrowđpcm\)
thanks