Question

check the roots of x-1/x=3 are irrational are not
​

Answer 1

Answer:

(x−1)(x−3)(x−5)(x−7)=9

The pattern

x−1,x−3,x−5,x−7

is significant. You can convert the ugly product

(x−1)(x−3)(x−5)(x−7)

into a symmetric product which can be expanded easily if you make a simple substitution.

First, find the mean of the four terms.

μ=(x−1)+(x−3)+(x−5)+(x−7)4

⟹μ=4x−164

⟹μ=x−4

⟹x=μ+4

Substitute for x in the original equation and see the magic.

[(μ+4)−1][(μ+4)−3][(μ+4)−5][(μ+4)−7]=9

⟹(μ+3)(μ+1)(μ−1)(μ−3)=9

Real pretty. Group the middle terms and the extreme two terms and expand them. That’s easy.

(μ2−9)(μ2−1)=9

⟹μ4−10μ2+9=9

⟹μ4−10μ2=0

⟹μ2(μ2−10)=0

Hence, the roots of this modified equation are given by the following quadruplet.

(0,0,10−−√,−10−−√)

However, x=μ+4, so that the roots of the original equation are given by this quadruplet.

(4,4,4+10−−√,4−10−−√)

The irrational roots being 4±10−−√, the sum of the irrational roots is  8

Step-by-step explanation: