# What is the discriminant of 4x^2-4x+11=0 and what does that mean?

Aug 5, 2015

$\Delta = - 160$

#### Explanation:

For a general form quadratic equation

$\textcolor{b l u e}{a {x}^{2} + b x + c = 0}$

the discriminant is defined as

$\textcolor{b l u e}{\Delta = {b}^{2} - 4 a c}$

$4 {x}^{2} - 4 x + 11 = 0$

which means that $a = 4$, $b = - 4$, and $c = 11$.

The discriminat will be equal to

$\Delta = {\left(- 4\right)}^{2} - 4 \cdot 4 \cdot 11$

$\Delta = 16 - 176 = \textcolor{g r e e n}{- 160}$

The fact that the discriminat is negative tells you that this quadratic has no real solutions, but that it does have two distinct imaginary roots.

Moreover, the graph of the function will have no $x$-intercept.

graph{4x^2 - 4x + 11 [-23.75, 27.55, 3.02, 28.68]}

The two roots will take the form

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

${x}_{1 , 2} = \frac{- \left(- 4\right) \pm \sqrt{- 160}}{2 \cdot 4} = \frac{4 \pm \sqrt{- 160}}{8}$
${x}_{1 , 2} = \frac{4 \pm 4 i \sqrt{10}}{8} = \left\{\begin{matrix}{x}_{1} = \frac{1 + i \sqrt{10}}{2} \\ {x}_{2} = \frac{1 - i \sqrt{10}}{2}\end{matrix}\right.$