Determine the real number A such that the following equation has a real answer for...

Cassarrim1

Cassarrim1

Answered

2022-02-01

Determine the real number A such that the following equation has a real answer for C for every real B:A2+B2+C2+2ABC=1

Answer & Explanation

sphwngzt

sphwngzt

Expert

2022-02-02Added 11 answers

Step 1
Solve for C:
C=AB±A2B2A2B2+1
We need a real A such that
A2B2A2B2+10
for any real B. Notice that we can factor this to get
(A21)(B21)0.
Since
B210 for all |B|1 and B21<0 for all |B|<1 there are only two A-values that can maintain the above inequality regardless of the B-value; that is,
A=±1
Souticexi

Souticexi

Expert

2022-02-03Added 6 answers

Step 1
Whichever variable we solve for, we discover the same thing.
A2+B2+C2+2ABC=1
A2+(2BC)A+(B2+C21)=0
A=2BC±(2BC)24(1)(B2+C21)2
A=BC±(B21)(C21)
Here, we can see that both variables under that radical must be greater than one of less than one. Since the variables are interchangeable, we know that all three must share this property so, either A, B, C1orA, B, C1

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