The-roots-of-the-equation-3-x-4-2-x-4-5-2x-4-are-a-all-real-b-all-imaginary-c-two-real-and-two-imaginary-d-none-of-the-above-

Question Number 20552 by ajfour last updated on 28/Aug/17

T h e r o o t s o f t h e e q u a t i o n

{(3 - x)}^{4} + {(2 - x)}^{4} = {(5 - 2 x)}^{4} a r e

(a) a l l r e a l (b) a l l i m a g i n a r y

(c) t w o r e a l a n d t w o i m a g i n a r y

(d) n o n e o f t h e a b o v e .

Answered by Tinkutara last updated on 28/Aug/17

{(x - 3)}^{4} + {(x - 2)}^{4} = {(2 x - 5)}^{4}

L e t a = x - 3, b = x - 2, t h e n s o l v i n g f o r

a^{4} + b^{4} = {(a + b)}^{4}

a b (2 a^{2} + 2 b^{2} + 3 a b) = 0

O n e s o l u t i o n i s a b = 0 \Rightarrow (x - 3) (x - 2) = 0

\Rightarrow x = 2, 3 (T w o i n t e g r a l r o o t s)

a n d 2 {(a + b)}^{2} = a b

\Rightarrow 2 {(2 x - 5)}^{2} = x^{2} - 5 x + 6

2 (4 x^{2} - 20 x + 25) = x^{2} - 5 x + 6

7 x^{2} - 35 x + 44 = 0

D = 1225 - 4 \times 7 \times 44 = - 7 < 0

H e n c e 2 r e a l a n d 2 i m a g i n a r y r o o t s .

Commented by ajfour last updated on 28/Aug/17

E x c e l l e n t! k e e p i t u p .

Facebook Tweet Pin