Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?

Question

The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is

(A)0

(B)2

(C)3

(d)4

Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot solve them,he told me post these questions on this site someone might know the answers and for now he is not in the town. So can you please help me.

Thank you.

Answer 1

HINT:

$(x-1)(3x-2)=3x^2-5x+2$ and $(x-2)(3x+1)=3x^2-5x-2$

Put $3x^2-5x=u$

Answer 2

Hint

$$(x-2)(3x+1)=3x^2-5x-2$$ $$(x-1)(3x-2)=3x^2-5x+2$$

If you denote $3x^2-5x$ by $y$ you get

$$(y-2)(y+2)=21$$

Answer 3

Hint : $$ (3x-2)(x-1) = 3x^2-5x+2 $$ and $$ (3x+1)(x-2) = 3x^2 -5x-2 $$

$$ (x-1)(x-2)(3x-2)(3x+1) = (3x^2-5x-2)(3x^2-5x+2) = 21 $$ Now put $$ 3x^2-5x = t $$ $$ (t-2)(t+2) = 21 $$ Now, solve