9x-1/2=27x2-1

Solution for 9x-1/2=27x2-1 equation:

9x-1/2=27x^2-1

We move all terms to the left:

9x-1/2-(27x^2-1)=0

We get rid of parentheses

-27x^2+9x+1-1/2=0

We multiply all the terms by the denominator


-27x^2*2+9x*2-1+1*2=0

We add all the numbers together, and all the variables

-27x^2*2+9x*2+1=0

Wy multiply elements

-54x^2+18x+1=0

a = -54; b = 18; c = +1;
Δ = b2-4ac
Δ = 182-4·(-54)·1
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{-(18)-6\sqrt{15}}{2*-54}=\frac{-18-6\sqrt{15}}{-108}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+6\sqrt{15}}{2*-54}=\frac{-18+6\sqrt{15}}{-108}
$