# How do you find the discriminant and how many solutions does 11x^2-9x-1=0? have?

May 1, 2015

The discriminant is the thing you take the square root of in the

$a {x}^{2} + b x + c = 0$ has solution(s): $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The discriminant is ${b}^{2} - 4 a c$

If $a , b$ and $c$ are real numbers, then:
If the discriminant is positive, then there are two real solutions.
$\textcolor{w h i t e}{\text{ssss}}$ One when we add and another when we subtract.

If the discriminant is 0, then there is one real solutions.
$\textcolor{w h i t e}{\text{ssss}}$ Since $\sqrt{0} = 0$, adding and subtracting do not give us different answers.

If the discriminant is positive, then there are two imaginary solutions.
$\textcolor{w h i t e}{\text{ssss}}$ Since the square root of a negative is imaginary, we get imaginary solutions.

In $11 {x}^{2} - 9 x - 1 = 0$, we have

$a = 11$, $b = - 9$ and $c = - 1$, so the discriminant is:

${b}^{2} - 4 a c = {\left(- 9\right)}^{2} - 4 \left(11\right) \left(- 1\right)$, which is equal to :

$81 + 44 = 125$

The equation has two real solutions.
$\textcolor{w h i t e}{\text{ssss}}$ One when we add $\sqrt{125}$ and another when we subtract $\sqrt{125}$.