9(x)=1/2x-3

Solution for 9(x)=1/2x-3 equation:

9(x)=1/2x-3

We move all terms to the left:

9(x)-(1/2x-3)=0


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

x∈R


We get rid of parentheses

9x-1/2x+3=0

We multiply all the terms by the denominator


9x*2x+3*2x-1=0

Wy multiply elements

18x^2+6x-1=0

a = 18; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·18·(-1)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-6\sqrt{3}}{2*18}=\frac{-6-6\sqrt{3}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+6\sqrt{3}}{2*18}=\frac{-6+6\sqrt{3}}{36}
$