4^x=3/2x+3

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

4^x=3/2x+3

We move all terms to the left:

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


Domain of the equation: 2x+3)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

8x^2-6x-3=0

a = 8; b = -6; c = -3;
Δ = b2-4ac
Δ = -62-4·8·(-3)
Δ = 132
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:

$\sqrt{\Delta}=\sqrt{132}=\sqrt{4*33}=\sqrt{4}*\sqrt{33}=2\sqrt{33}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{33}}{2*8}=\frac{6-2\sqrt{33}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{33}}{2*8}=\frac{6+2\sqrt{33}}{16}
$