A Polynomial

Algebra Level 2

If, ( x − 9 ) 2 + ( y − 4 ) 2 + ( z − 8 ) 2 + ( w − 7 ) 2 = 0 (x-9)^2 + (y-4)^2 + (z-8)^2 + (w-7)^2 = 0 Then x y z w + 1 3 2 xyzw+13^2 is equal to:


The answer is 2185.

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4 solutions

David Watkins
Aug 1, 2016

Geometrically, this equation represents a hypersphere with centre (9, 4, 8, 7) and radius 0 in 4-dimensional space.

Since the radius of the hypersphere is 0, it means that (9, 4, 8, 7) is the only possible point in the hypersphere. Thus x = 9, y = 4, z = 8 and w = 7 is the only solution to the equation, and so

x y z w + 1 3 2 = 9 ∗ 4 ∗ 8 ∗ 7 + 169 = 2185 xyzw + 13^2 = 9 * 4 * 8 * 7 + 169 = 2185

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Zainab Tariq
Mar 22, 2014

very simple... if the answer to an addition question is zero then it means that it was zero that was added all along...:)

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2185

Maya Patil - 7 years, 2 months ago
Damiann Mangan
Mar 17, 2014

Easily, ( x , y , z , w ) = ( 9 , 4 , 8 , 7 ) (x,y,z,w) = (9,4,8,7) (assuming each and every variables are rational) as a square never goes negative.

This leads us to x y z w + 1 3 2 = 2016 + 169 = 2185 xyzw + 13^{2} = 2016 + 169 = 2185 as the only solution .

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correct

Maya Patil - 7 years, 2 months ago

This is only possible when all of this terms are zero because this term are square and can be either 0 or positive. Therefore, x=9,y=4,z=8, w=7.

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