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Let a, b, c, and d be distinct real numbers such that

a=4+5+a,b=45+b,c=4+5c,d=45d.
Compute abcd.

 Jan 28, 2021
edited by Melody  May 11, 2021

Best Answer 

 #3
avatar+87 
+9

 

Squaring both sides of the original equation, we get a2=4+5+a.
Subtracting 4 from both sides and squaring again gives (a24)2=5+a.
Expanding this out and subtracting 5 + a from both sides, we have a48a2a+11=0.
Similar manipulations on the other equations give b48b2b+11=0,c48c2+c+11=0,d48d2+d+11=0.
(Note the signs carefully.)

Let f(x)=x48x2x+11. Then a and b are roots of f(x), while c and d are roots of  x48x2+x+11, which is f(-x). It follows that -c and -d are roots of f(x). Since -c and -d are negative and hence distinct from a and b, we have four roots of f(x). Since f(x) has degree four, we know we have found all the roots.

By Vieta's formulas, the constant term of f(x) is the product of the roots. Thus, 

 

abcd=(a)(b)(c)(d)=11.

 

laugh

 Apr 16, 2021
edited by Divineology  Apr 16, 2021
 #1
avatar
-6

The solutions are 2.2192, 1.33938, 1.09044, and 2.4682, and their product is 8.

 Jan 28, 2021
 #2
avatar+118703 
+2

I'd like to see this one answered.

 

Divineology, perhaps you can show us how it is done?    Or anyone else?  

 Apr 15, 2021
 #3
avatar+87 
+9
Best Answer

 

Squaring both sides of the original equation, we get a2=4+5+a.
Subtracting 4 from both sides and squaring again gives (a24)2=5+a.
Expanding this out and subtracting 5 + a from both sides, we have a48a2a+11=0.
Similar manipulations on the other equations give b48b2b+11=0,c48c2+c+11=0,d48d2+d+11=0.
(Note the signs carefully.)

Let f(x)=x48x2x+11. Then a and b are roots of f(x), while c and d are roots of  x48x2+x+11, which is f(-x). It follows that -c and -d are roots of f(x). Since -c and -d are negative and hence distinct from a and b, we have four roots of f(x). Since f(x) has degree four, we know we have found all the roots.

By Vieta's formulas, the constant term of f(x) is the product of the roots. Thus, 

 

abcd=(a)(b)(c)(d)=11.

 

laugh

Divineology  Apr 16, 2021
edited by Divineology  Apr 16, 2021
 #4
avatar+118703 
+1

Diveneology:

Thank you very much for your answer.    laughlaughlaugh

 Apr 16, 2021

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