Question

The number of real roots of (3-x)^4 + (5-x)^4 = 16 is?

Answer 1

Expand the first side and let what you get =16

Answer 2

the answer should be 2...

Answer 3

\[
(3-x)^4=x^4-12 x^3+54 x^2-108 x+81\\
(5-x)^4 =x^4-20 x^3+150 x^2-500 x+625\\
\]
Now add both and set equal to 16

Answer 4

Actually x=3 is a solution.

Answer 5

and x=5 is also a solution.

Answer 6

There are two real solutions 3 and 5 and two complex solutions\[
4 \pm i \sqrt 7
\]

Answer 7

\[
(3-x)^4+(5-x)^4-16=\\2 x^4-32
x^3+204 x^2-608 x+690=\\2
(x-5) (x-3) \left(x^2-8
x+23\right)
\]
Since 5 and 3 were easily found, you can divide by (x-5)(x-3) and get the factorization above.

Answer 8

The answer to the problem is 2.