perplexus.info :: Just Math : Nested Radical Product

Nested Radical Product (Posted on 2025-03-05)

If

a = √(4 - √(5 - a))
b = √(4 + √(5 - b))
c = √(4 - √(5 + c))
d = √(4 + √(5 + d))

Determine the value of the product abcd.

No Solution Yet Submitted by Danish Ahmed Khan

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no computer

Comment 5 of 5 |

If you square, rearrange, and square again, the first two equations take the form f(x)=x^4-8x^2+x+11 and the last two take the form

g(x)=x^4-8x^2-x+11.

a and b are roots of f(x) and c and d are roots of g(x). But f(-c)=g(c) so that -c is also a root of f(x), and likewise -d.

Then f(x)=(x-a)(x-b)(x+c)(x+d) with constant term abcd=11.

Posted by xdog on 2025-03-06 10:25:55

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