4^x=3/2x+6

Solution for 4^x=3/2x+6 equation:

4^x=3/2x+6

We move all terms to the left:

4^x-(3/2x+6)=0


Domain of the equation: 2x+6)!=0

x∈R


We get rid of parentheses

4^x-3/2x-6=0

We multiply all the terms by the denominator


4^x*2x-6*2x-3=0

Wy multiply elements

8x^2-12x-3=0

a = 8; b = -12; c = -3;
Δ = b2-4ac
Δ = -122-4·8·(-3)
Δ = 240
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:

$\sqrt{\Delta}=\sqrt{240}=\sqrt{16*15}=\sqrt{16}*\sqrt{15}=4\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-4\sqrt{15}}{2*8}=\frac{12-4\sqrt{15}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+4\sqrt{15}}{2*8}=\frac{12+4\sqrt{15}}{16}
$