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`        1.Find the minimum value of3x^2+5y^2+4z^2-4xy-4xz-4x-2y+25 for real numbers x,y and z.2.suppose a,b,c are real numbers such that ('a' is a negative real number) ab-a=b+119, bc-b=c+59, ca-c=a+71. Find a+b+c`
one year ago

```								We would try to make whole squares to get minimum value Rewrite as(x-2y)2 +(x-2z)2+(x-2)2+(y-1)2+20squares of any number >=0 So minimum value of expression =20 2.  ab-a=b+119 , bc-b=c+59 , ca-c =a+71ab-a-b=119 , bc-b-c=59, ca-c-a=71Adding 1 on both sides we getab-a-b+1=120 a(b-1)-(b-1)=120(a-1)(b-1)=120 ----(1) similarly adding 1 to both side in other two equation we will get (b-1)(c-1)=60 -----(2) (a-1)(c-1)=72 -----(3) divide 1 by 2 a-1/c-1=2 and from 3  putting value of (c-1)(a-1)2/72=2(a-1)2=144a-1=-12 (a is negative )a=-11Similarlyb=-9c=-5
```
10 months ago
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