Forwarded from Math World (Azizakhon Turakhujaeva)
Forwarded from Deleted Account
=>
a^2+b^2+c^2+2(ab+bc+ac)=4^2
ab+bc+ac=3
(a+b+c)(a^2+b^2+c^2)=40
22+4(ab+ac+bc)-3abc=40
3abc=-6, abc=-2
(ab+bc+ac)^2=9
(ab)^2+(bc)^2+(ac)^2=9-2•4•(-2)=25
(a^2+b^2+c^2)^2=100
a^4+b^4+c^4=100-2•25=50
a^2+b^2+c^2+2(ab+bc+ac)=4^2
ab+bc+ac=3
(a+b+c)(a^2+b^2+c^2)=40
22+4(ab+ac+bc)-3abc=40
3abc=-6, abc=-2
(ab+bc+ac)^2=9
(ab)^2+(bc)^2+(ac)^2=9-2•4•(-2)=25
(a^2+b^2+c^2)^2=100
a^4+b^4+c^4=100-2•25=50
Forwarded from Yogesh
4x+5ky+7=0
Kx-6y+12=0
X/60k+42=y/7k-48=1/-24-5k^2
In 2nd quadrant x<0 y>0
60k+42/-24-5k^2)<0
60k+42/24+5k^2>0
But 24+5k^2>0. Ignore it
It will be positive when
60k+42>0
K>-7/10
Y=7k-48/-24-5k^2. >0
7k-48<0
K<48/7
So. -7/10<k<48/7
0,1,2,3,4,5,6. 7 values
Kx-6y+12=0
X/60k+42=y/7k-48=1/-24-5k^2
In 2nd quadrant x<0 y>0
60k+42/-24-5k^2)<0
60k+42/24+5k^2>0
But 24+5k^2>0. Ignore it
It will be positive when
60k+42>0
K>-7/10
Y=7k-48/-24-5k^2. >0
7k-48<0
K<48/7
So. -7/10<k<48/7
0,1,2,3,4,5,6. 7 values
Forwarded from Math World (Azizakhon Turakhujaeva)
x^5+1/x^5=205/16•(x+1/x)
x=a, 1/x=b
a^5+b^5=(a+b)^5-5a^4•b-10a^3•b^2-10a^2•b^3-5a•b^4=(a+b)^5-5ab(a^3+b^3)-10a^2•b^2(a+b)=
(a+b)^5-5((a+b)^3-3(a+b))-10(a+b)
a^5+b^5=(a+b)^5-5(a+b)^3+5(a+b)
x^5+1/x^5=205/16•(x+1/x)
(x+1/x)^4-5(x+1/x)^2+5=205/16
(x+1/x)^2=y
16y^2-80y-125=0
y=6.25
(x+1/x)^2=6.25
x+1/x=5/2
2x^2-5x+2=0
x=2; 1/2
x=a, 1/x=b
a^5+b^5=(a+b)^5-5a^4•b-10a^3•b^2-10a^2•b^3-5a•b^4=(a+b)^5-5ab(a^3+b^3)-10a^2•b^2(a+b)=
(a+b)^5-5((a+b)^3-3(a+b))-10(a+b)
a^5+b^5=(a+b)^5-5(a+b)^3+5(a+b)
x^5+1/x^5=205/16•(x+1/x)
(x+1/x)^4-5(x+1/x)^2+5=205/16
(x+1/x)^2=y
16y^2-80y-125=0
y=6.25
(x+1/x)^2=6.25
x+1/x=5/2
2x^2-5x+2=0
x=2; 1/2
Forwarded from Math World (Azizakhon Turakhujaeva)
x^6-7x^5+10x^4-8x^3+10x^2-7x+1=0
x^6-7x^5+6x^4+4x^4-8x^3+10x^2-7x+1=0
x^4(x-6)(x-1)+4x^3(x-1)-
4x^2(x-1)+6x(x-1)-(x-1)=0
(x-1)•(x^3-6x^4+4x^3-4x^2+6x-1)=0
x1=1
x^3-6x^4+4x^3-4x^2+6x-1=0
x^3-1-6x(x^2-1)+4x^2(x-1)=0
(x-1)(x^2+x+1-6x^2-6x+4x^2)=0
-x^2-5x+1=0
x^2+5x-1=0
D=25+4=29
x2=(-5+✔️29)/2
x3=(-5-✔️29)/2
x^6-7x^5+6x^4+4x^4-8x^3+10x^2-7x+1=0
x^4(x-6)(x-1)+4x^3(x-1)-
4x^2(x-1)+6x(x-1)-(x-1)=0
(x-1)•(x^3-6x^4+4x^3-4x^2+6x-1)=0
x1=1
x^3-6x^4+4x^3-4x^2+6x-1=0
x^3-1-6x(x^2-1)+4x^2(x-1)=0
(x-1)(x^2+x+1-6x^2-6x+4x^2)=0
-x^2-5x+1=0
x^2+5x-1=0
D=25+4=29
x2=(-5+✔️29)/2
x3=(-5-✔️29)/2