x 2 + x + = 0

### Result:

Nature of the roots:
Root1:
Root2:

Quadratic equation is a 2nd degree polynomial euqtion, generally represented as a x 2 + b x + c = 0, where a ≠ 0. Let's calculate the roots of the equation and nature of the roots.
a x 2 + b x + c = 0; where a ≠ 0
on divinding with 'a' on both sides we get
x 2 + $\frac{{b}^{}}{{a}^{}}$ x + $\frac{{c}^{}}{{a}^{}}$ = 0
x 2 + 2 . $\frac{{b}^{}}{{\mathrm{2a}}^{}}$ x + $\frac{{c}^{}}{{a}^{}}$ = 0
let's add and subtract $\frac{{b}^{2}}{{\mathrm{4a}}^{2}}$ to form a perfect squre with the first two terms
x 2 + 2 . $\frac{{b}^{}}{{\mathrm{2a}}^{}}$ x + $\frac{{b}^{2}}{{\mathrm{4a}}^{2}}$ - $\frac{{b}^{2}}{{\mathrm{4a}}^{2}}$ + $\frac{{c}^{}}{{a}^{}}$ = 0
using (a+b)2 = a 2 + 2.a.b. + b 2 formula
( x + $\frac{{b}^{}}{{\mathrm{2a}}^{}}$) 2 + $\frac{{c}^{}}{{a}^{}}$ - $\frac{{b}^{2}}{{\mathrm{4a}}^{2}}$ + = 0
root1 = $\frac{{-b}^{-}}{{\mathrm{2a}}^{}}$ and root2 = $\frac{{-b}^{+}}{{\mathrm{2a}}^{}}$
where D = b 2 - 4ac is called discriminant.
If D > 0 then the equationhas two distict real roots mentioned above.
If D=0 then two equal roots.
If D < 0 then roots are complex numbers

This algebraic equation has application in almost all the branches of science in one form or the other, and very basic equation. another way of looking at the roots of the equation is as follows.
Let p and q be the roots of the equation a x 2 + b x + c = 0, where a ≠ 0.
Then sum of the roots p + q = - $\frac{{b}^{}}{{a}^{}}$ and product of the roots is pq = $\frac{{c}^{}}{{a}^{}}$