x*12^x=84

Solution for x*12^x=84 equation:

x*12^x=84

We move all terms to the left:

x*12^x-(84)=0

Wy multiply elements

12x^2-84=0

a = 12; b = 0; c = -84;
Δ = b2-4ac
Δ = 02-4·12·(-84)
Δ = 4032
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{4032}=\sqrt{576*7}=\sqrt{576}*\sqrt{7}=24\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-24\sqrt{7}}{2*12}=\frac{0-24\sqrt{7}}{24}
=-\frac{24\sqrt{7}}{24}
=-\sqrt{7}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+24\sqrt{7}}{2*12}=\frac{0+24\sqrt{7}}{24}
=\frac{24\sqrt{7}}{24}
=\sqrt{7}
$