Double root

The double root instance occurs when a quadratic equation is the square of a binomial, or when its discriminant is equal to $0$ .

Example

Find all solutions to the quadratic equation $3x^2+6x+3=0$ .

Plugging the values of $a, b$ and $c$ into the quadratic formula, we get

$\frac{-6\pm\sqrt{6^2-4(3)(3)}}{2(3)}=\frac{-6\pm\sqrt{36-36}}{6}=\frac{-6\pm\sqrt{0}}{6}=\frac{-6}{6}=-1$

so this quadratic has a double root of $-1$ .

We also could have solved this problem by factoring a $3$ out of the left side and dividing:

$3(x^2+2x+1)=0 \Longrightarrow x^2+2x+1=0$

and now we plug the coefficients into the quadratic formula:

$\frac{-2\pm\sqrt{2^2-4(1)(1)}}{2(1)}=\frac{-2\pm\sqrt{4-4}}{2}=\frac{-2}{2}=-1$

so again, the quadratic has a double root of $-1$ .