MHB Solve $(2x+1)(3x+1)(5x+1)(30x+1)=10$: Real Solutions

[anemone]

Find all real solutions of the equation $(2x+1)(3x+1)(5x+1)(30x+1)=10$.

 

[kaliprasad]

Find all real solutions of the equation $(2x+1)(3x+1)(5x+1)(30x+1)=10$.

Spoiler

we have $(2x+1)(30x+1)(3x+1)(5x+1) = 10$
or $(60x^2+ 32x + 1)(15x^2+ 8x + 1) = 10$

letting $15x^2 + 8x = t$

$(4t+1)(t+1) = 10$

or $4t^2 + 5 t + 1 = 10$

or $4t^2 + 5t - 9 = 0$

or $(4t+9)(t-1) =0 $



t = 1 or -9/4

t = 1 gives

$15x^2 + 8x-1=0$ giving $x = \dfrac{-4\pm\sqrt{31}}{15}$or $(15x^2 + 8x +\frac{9}4{4}) = 0$

or $(60x^2+ 32x + 9) = 0$

this gives complex solution

so solutions are $x = \dfrac{-4\pm\sqrt{31}}{15}$