16^x=1/4^x-5

Solution for 16^x=1/4^x-5 equation:

16^x=1/4^x-5

We move all terms to the left:

16^x-(1/4^x-5)=0


Domain of the equation: 4^x-5)!=0

x∈R


We get rid of parentheses

16^x-1/4^x+5=0

We multiply all the terms by the denominator


16^x*4^x+5*4^x-1=0

Wy multiply elements

64x^2+20x-1=0

a = 64; b = 20; c = -1;
Δ = b2-4ac
Δ = 202-4·64·(-1)
Δ = 656
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:

$\sqrt{\Delta}=\sqrt{656}=\sqrt{16*41}=\sqrt{16}*\sqrt{41}=4\sqrt{41}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-4\sqrt{41}}{2*64}=\frac{-20-4\sqrt{41}}{128}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+4\sqrt{41}}{2*64}=\frac{-20+4\sqrt{41}}{128}
$