Question

What is the sum of the irrational roots of the equation (x-1)(x-3)(x-5)(x-7)=9 ?

Answer 1

Let x - 4 = p
Then the given eqn becomes

(p + 3) (p + 1) (p - 1) (p - 3) = 9

(p2 - 1) (p2 - 9) = 9

p4 - 10p2 + 9 = 9

p2 (p2 - 10) = 0

p2 =0 or p2-10 =0

p = 0 or p =  \sqrt{10}  or p = -  \sqrt{10} 

then x - 4 = 0, x - 4 =  \sqrt{10} or x - 4 = -  \sqrt{10} 

Now the roots of the given eqn are 4,4 +  \sqrt{10}  and 4 -  \sqrt{10} 

The irrational roots are 4+ \sqrt{10}  and 4 -  \sqrt{10} 

The sum of the irrational roots = 4 +  \sqrt{10}  + 4 -  \sqrt{10}  = 8.

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Answer 2

Answer:

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