# What is the discriminant of 9x^2-6x+1=0 and what does that mean?

Jul 27, 2015

For this quadratic, $\Delta = 0$, which means that the equation has one real root (a repeated root).

#### Explanation:

The general form of a quadratic equation looks like this

$a {x}^{2} + b x + c = 0$

The discriminant of a quadratic equation is defined as

$\Delta = {b}^{2} - 4 \cdot a \cdot c$

In your case, the equation looks like this

$9 {x}^{2} - 6 x + 1 = 0$,

which means that you have

$\left\{\begin{matrix}a = 9 \\ b = - 6 \\ c = 1\end{matrix}\right.$

The discriminant will thus be equal to

$\Delta = {\left(- 6\right)}^{2} - 4 \cdot 9 \cdot 1$

$\Delta = 36 - 36 = \textcolor{g r e e n}{0}$

When the discrimiannt is equal to zero, the quadratic will only have one distinct real solution, derived from the general form

${x}_{1 , 2} = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{- 6 \pm \sqrt{0}}{2 a} = \textcolor{b l u e}{- \frac{b}{2 a}}$

In your case, the equation has one distinct real solution equal to

${x}_{1} = {x}_{2} = - \frac{\left(- 6\right)}{2 \cdot 9} = \frac{6}{18} = \frac{1}{3}$