cho a+b+c=0, a^2+b^2+c^2=14. Tính a^4+b^4+c^4
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x3 - 5x2 + 8x - 4
= x3 - x2 - 4x2 + 4x + 4x - 4
= x2( x - 1) - 4x (x - 1) + 4 (x - 1)
= (x2 - 4x + 4) (x - 1)
= (x - 2)2 (x - 1)
x - 1 = 0
x = 1
(x - 2)2 = 0
x = 2
vậy x = 1
x = 2
Ta thấy:A=\(\frac{10^{19}+1}{10^{20}+1}\)=>10A=\(\frac{10^{20}+10}{10^{20}+1}\)
=>10A=\(\frac{10^{20}+1+9}{10^{20}+1}\)
=>10A=1+\(\frac{9}{10^{20}+1}\)
Ta thấy:B=\(\frac{10^{20}+1}{10^{21}+1}\)
=>10B=\(\frac{10^{21}+10}{10^{21}+1}\)
=>10B=\(\frac{10^{21}+1+9}{10^{21}+1}\)
=>10B=1+\(\frac{9}{10^{21}+1}\)
Do \(\frac{9}{10^{20}+1}\)> \(\frac{9}{10^{21}+1}\)=>A > B
Áp dụng Bđt Cauchy-schwarz ta có:
\(\left(1^2+1^2+1^2\right)\left(x^2+y^2+z^2\right)\ge\left(x+y+z\right)^2=3^2=9\)
\(\Rightarrow3\left(x^2+y^2+z^2\right)\ge9\)
\(\Rightarrow x^2+y^2+z^2\ge3\)
\(\Rightarrow P\ge3\)
Dấu = khi x=y=z=1
a+b+c = 0
<=> (a+b+c)^2 = 0
<=> a^2 + b^2 + c^2 + 2 ab + 2ac + 2bc = 0
<=>14 + 2(ab + ac + bc) = 0
<=> 2(ab + ac + bc) = -14
<=> ab + ac + bc = -7
=> (ab + ac + bc)^2 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2a^2bc + 2 ab^2c + 2abc^2 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2abc(a + b + c) = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2abc . 0 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 = 49
Ta có: a^2 + b^2 + c^2 = 14
=> (a^2 + b^2 + c^2)^2 = 14^2
<=> a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2 b^2c^2 =196
<=> a^4 + b^4 + c^4 + 2(a^2b^2 + a^2c^2 + b^2c^2) = 196
<=> a^4 + b^4 + c^4 + 2 . 49 = 196
<=> a^4 + b^4 + c^4 + 98 = 196
<=> a^4 + b^4 + c^4 = 98
a+b+c = 0
<=> (a+b+c)^2 = 0
<=> a^2 + b^2 + c^2 + 2 ab + 2ac + 2bc = 0
<=>14 + 2(ab + ac + bc) = 0
<=> 2(ab + ac + bc) = -14
<=> ab + ac + bc = -7
=> (ab + ac + bc)^2 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2a^2bc + 2 ab^2c + 2abc^2 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2abc(a + b + c) = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 + 2abc . 0 = 49
<=> a^2b^2 + a^2c^2 + b^2c^2 = 49
Ta có: a^2 + b^2 + c^2 = 14
=> (a^2 + b^2 + c^2)^2 = 14^2
<=> a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2 b^2c^2 =196
<=> a^4 + b^4 + c^4 + 2(a^2b^2 + a^2c^2 + b^2c^2) = 196
<=> a^4 + b^4 + c^4 + 2 . 49 = 196
<=> a^4 + b^4 + c^4 + 98 = 196
<=> a^4 + b^4 + c^4 = 98