3x^2=-4.5x-1

Solution for 3x^2=-4.5x-1 equation:

3x^2=-4.5x-1

We move all terms to the left:

3x^2-(-4.5x-1)=0

We get rid of parentheses

3x^2+4.5x+1=0

a = 3; b = 4.5; c = +1;
Δ = b2-4ac
Δ = 4.52-4·3·1
Δ = 8.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4.5)-\sqrt{8.25}}{2*3}=\frac{-4.5-\sqrt{8.25}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4.5)+\sqrt{8.25}}{2*3}=\frac{-4.5+\sqrt{8.25}}{6}
$