X^2-5=2x+1/3

Solution for X^2-5=2x+1/3 equation:

X^2-5=2X+1/3

We move all terms to the left:

X^2-5-(2X+1/3)=0

We add all the numbers together, and all the variables

X^2-(+2X+1/3)-5=0

We get rid of parentheses

X^2-2X-5-1/3=0

We multiply all the terms by the denominator


X^2*3-2X*3-1-5*3=0

We add all the numbers together, and all the variables

X^2*3-2X*3-16=0

Wy multiply elements

3X^2-6X-16=0

a = 3; b = -6; c = -16;
Δ = b2-4ac
Δ = -62-4·3·(-16)
Δ = 228
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:

$\sqrt{\Delta}=\sqrt{228}=\sqrt{4*57}=\sqrt{4}*\sqrt{57}=2\sqrt{57}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{57}}{2*3}=\frac{6-2\sqrt{57}}{6}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{57}}{2*3}=\frac{6+2\sqrt{57}}{6}
$