2^x*x-x^2=135

Solution for 2^x*x-x^2=135 equation:

2^x*x-x^2=135

We move all terms to the left:

2^x*x-x^2-(135)=0

We add all the numbers together, and all the variables

-1x^2+2^x*x-135=0

Wy multiply elements

-1x^2+2x^2-135=0

We add all the numbers together, and all the variables

x^2-135=0

a = 1; b = 0; c = -135;
Δ = b2-4ac
Δ = 02-4·1·(-135)
Δ = 540
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:

$\sqrt{\Delta}=\sqrt{540}=\sqrt{36*15}=\sqrt{36}*\sqrt{15}=6\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{15}}{2*1}=\frac{0-6\sqrt{15}}{2}
=-\frac{6\sqrt{15}}{2}
=-3\sqrt{15}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{15}}{2*1}=\frac{0+6\sqrt{15}}{2}
=\frac{6\sqrt{15}}{2}
=3\sqrt{15}
$