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1/Ta có: \(\left(a+b+c\right)^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)=81\)
\(\Rightarrow M=ab+bc+ca=\frac{\left(81-141\right)}{2}\)
Bài 1.
A = x2 + 2xy + y2 = ( x + y )2 = ( -1 )2 = 1
B = x2 + y2 = ( x2 + 2xy + y2 ) - 2xy = ( x + y )2 - 2xy = (-1)2 - 2.(-12) = 1 + 24 = 25
C = x3 + 3xy( x + y ) + y3 = ( x3 + y3 ) + 3xy( x + y ) = ( x + y )( x2 - xy + y2 ) + 3xy( x + y )
= -1( 25 + 12 ) + 3.(-12).(-1)
= -37 + 36
= -1
D = x3 + y3 = ( x3 + 3x2y + 3xy2 + y3 ) - 3x2y - 3xy2 = ( x + y )3 - 3xy( x + y ) = (-1)3 - 3.(-12).(-1) = -1 - 36 = -37
Bài 2.
M = 3( x2 + y2 ) - 2( x3 + y3 )
= 3( x2 + y2 ) - 2( x + y )( x2 - xy + y2 )
= 3( x2 + y2 ) - 2( x2 - xy + y2 )
= 3x2 + 3y2 - 2x2 + 2xy - 2y2
= x2 + 2xy + y2
= ( x + y )2 = 12 = 1
a, A = 27x3 + 54x2 + 36x + 4
=> A = (3x)3 + 2 . 3 . 9x2 + 3 . 3x . 4 + 8 - 4
=> A = [(3x)3 + 2 . 3 . (3x)2 + 3 . 3x . 22 + 23] - 4
=> A = (3x + 2)3 - 4
Thay x = -2002 vào A ta có: A = (3x + 2)3 - 4
=> A = [3 x (-2002) + 2]3 - 4 = [6006 + 2]3 - 4 = 60083 - 4
b, B = 2x2 + 4x + xy + 2y
=> B = 2x(x + 2) + y(x + 2)
=> B = (x + 2)(2x + y)
Thay x = 88; y = -76 vào B
=> B = (88 + 2)[2 . 88 + (-76)]
=> B = 90 . [176 + (-76)] = 90 . 100 = 9000
\(a\left(3x-1\right)\left(2x+7\right)-\left(x+1\right)\left(6x-5\right)-\left(18x-12\right)\)
\(=6x^2+21x-2x-7-\left(6x^2-5x+6x-5\right)-18x+12\)
\(=6x^2+21x-2x-7-6x^2+5x-6x-5-18x+12\)
\(=0\left(đpcm\right)\)
\(b,\left(x-y\right)\left(x^3+x^2y+xy^2+y^3\right)-x^4+y^4\)
\(=x^4+x^3y+x^2y^2+xy^3-x^3y-x^2y^2-xy^3-y^4-x^4+y^4\)
\(=0\left(đpcm\right)\)
x+y=9=>x=9-y thay x=9-y vào xy=14
=> (9-y).y=14
=> 9y-y^2 -14=0
=> 7y+2y-y^2-14=0
=> y(7-y)-2(7-y)=0
=>(7-y)(y-2)=0
=>y=2 hoặc 7
=> x=7 hoặc 2
=> (x;y)=(2,7);(7,2)
a.
x-y = 7-2 = 5
hoặc 2-7=-5
b. x^2+y^2= 7^2+2^2 =53
hoặc 2^2+7^2=53
c. x^3+y^3 = 7^3 +2^3 = 351
hoặc 2^3 +7^3 =351
d.x^4+y^4= 7^4 +2^4= 2417
hoặc 2^4+7^4=2417
2a) \(4x^2-1=\left(2x\right)^2-1^2=\left(2x+1\right)\left(2x-1\right)\)
b) \(x^2+16x+64=\left(x+8\right)^2\)
c) \(x^3-8y^3=x^3-\left(2y\right)^3\)
\(=\left(x-2y\right)\left(x^2+2xy+4y^2\right)\)
d) \(9x^2-12xy+4y^2=\left(3x-2y\right)^2\)
1/ Ta có : P\left(x\right)=-x^2+13x+2012=-\left(x-\frac{13}{2}\right)^2+\frac{8217}{4}\le\frac{8217}{4}P(x)=−x2+13x+2012=−(x−213)2+48217≤48217
Dấu "=" xảy ra khi x = 13/2
Vậy Max P(x) = 8217/4 tại x = 13/2
1/ Ta có : P\left(x\right)=-x^2+13x+2012=-\left(x-\frac{13}{2}\right)^2+\frac{8217}{4}\le\frac{8217}{4}P(x)=−x2+13x+2012=−(x−213)2+48217≤48217
Dấu "=" xảy ra khi x = 13/2
Vậy Max P(x) = 8217/4 tại x = 13/2
2/ Ta có : x^3+3xy+y^3=x^3+3xy.1+y^3=x^3+y^3+3xy\left(x+y\right)=\left(x+y\right)^3=1x3+3xy+y3=x3+3xy.1+y3=x3+y3+3xy(x+y)=(x+y)3=1
3/ a+b+c=0\Leftrightarrow\left(a+b+c\right)^2=0\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ac\right)=0a+b+c=0⇔(a+b+c)2=0⇔a2+b2+c2+2(ab+bc+ac)=0
\Leftrightarrow ab+bc+ac=-\frac{1}{2}⇔ab+bc+ac=−21 \Leftrightarrow\left(ab+bc+ac\right)^2=\frac{1}{4}\Leftrightarrow a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)=\frac{1}{4}⇔(ab+bc+ac)2=41⇔a2b2+b2c2+c2a2+2abc(a+b+c)=41
\Leftrightarrow a^2b^2+b^2c^2+c^2a^2=\frac{1}{4}⇔a2b2+b2c2+c2a2=41(vì a+b+c=0)
Ta có : a^2+b^2+c^2=1\Leftrightarrow\left(a^2+b^2+c^2\right)^2=1\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=1a2+b2+c2=1⇔(a2+b2+c2)2=1⇔a4+b4+c4+2(a2b2+b2c2+c2a2)=1
\Leftrightarrow a^4+b^4+c^4=1-2\left(a^2b^2+b^2c^2+c^2a^2\right)=1-\frac{2.1}{4}=\frac{1}{2}⇔a4+b4+c4=1−2(a2b2+b2c2+c2a2)=1−42.1=21
(x - 5)² = (3 + 2x)²
(x - 5)² - (3 + 2x)² = 0
[(x - 5) - (3 + 2x)][(x - 5) + (3 + 2x)] = 0
(x - 5 - 3 - 2x)(x - 5 + 3 + 2x) = 0
(-x - 8)(3x - 2) = 0
-x - 8 = 0 hoặc 3x - 2 = 0
*) -x - 8 = 0
-x = 8
x = -8
*) 3x - 2 = 0
3x = 2
x = 2/3
Vậy x = -8; x = 2/3
--------------------
27x³ - 54x² + 36x = 9
27x³ - 54x² + 36x - 9 = 0
27x³ - 27x² - 27x² + 27x + 9x - 9 = 0
(27x³ - 27x²) - (27x² - 27x) + (9x - 9) = 0
27x²(x - 1) - 27x(x - 1) + 9(x - 1) = 0
(x - 1)(27x² - 27x + 9) = 0
x - 1 = 0 hoặc 27x² - 27x + 9 = 0
*) x - 1 = 0
x = 1
*) 27x² - 27x + 9 = 0
Ta có:
27x² - 27x + 9
= 27(x² - x + 1/3)
= 27(x² - 2.x.1/2 + 1/4 + 1/12)
= 27[(x - 1/2)² + 1/12] > 0 với mọi x ∈ R
⇒ 27x² - 27x + 9 = 0 (vô lí)
Vậy x = 1
A = x² + y²
= x² - 2xy + y² + 2xy
= (x - y)² + 2xy
= 4² + 2.1
= 16 + 2
= 18
B = x³ - y³
= (x - y)(x² + xy + y²)
= (x - y)(x² - 2xy + y² + xy + 2xy)
= (x - y)[(x - y)² + 3xy]
= 4.(4² + 3.1)
= 4.(16 + 3)
= 4.19
= 76
C = x⁴ + y⁴
= (x²)² + (y²)²
= (x²)² + 2x²y² + (y²)² - 2x²y²
= (x² + y²)² - 2x²y²
= (x² - 2x²y² + y² + 2x²y²)² - 2x²y²
= [(x - y)² + 2x²y²]² - 2x²y²
= (4² + 2.1²)² - 2.1²
= (16 + 2)² - 2
= 18² - 2
= 324 - 2
= 322