27(x)=1/3x+13

Solution for 27(x)=1/3x+13 equation:

27(x)=1/3x+13

We move all terms to the left:

27(x)-(1/3x+13)=0


Domain of the equation: 3x+13)!=0

x∈R


We get rid of parentheses

27x-1/3x-13=0

We multiply all the terms by the denominator


27x*3x-13*3x-1=0

Wy multiply elements

81x^2-39x-1=0

a = 81; b = -39; c = -1;
Δ = b2-4ac
Δ = -392-4·81·(-1)
Δ = 1845
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:

$\sqrt{\Delta}=\sqrt{1845}=\sqrt{9*205}=\sqrt{9}*\sqrt{205}=3\sqrt{205}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-39)-3\sqrt{205}}{2*81}=\frac{39-3\sqrt{205}}{162}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-39)+3\sqrt{205}}{2*81}=\frac{39+3\sqrt{205}}{162}
$