4^x=1/2^x-9

Solution for 4^x=1/2^x-9 equation:

4^x=1/2^x-9

We move all terms to the left:

4^x-(1/2^x-9)=0


Domain of the equation: 2^x-9)!=0

x∈R


We get rid of parentheses

4^x-1/2^x+9=0

We multiply all the terms by the denominator


4^x*2^x+9*2^x-1=0

Wy multiply elements

8x^2+18x-1=0

a = 8; b = 18; c = -1;
Δ = b2-4ac
Δ = 182-4·8·(-1)
Δ = 356
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:

$\sqrt{\Delta}=\sqrt{356}=\sqrt{4*89}=\sqrt{4}*\sqrt{89}=2\sqrt{89}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{89}}{2*8}=\frac{-18-2\sqrt{89}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{89}}{2*8}=\frac{-18+2\sqrt{89}}{16}
$