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

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

In your case, you have

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

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

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

In your case, you have

${x}_{1 , 2} = \frac{- \left(- 4\right) \pm \sqrt{- 160}}{2 \cdot 4} = \frac{4 \pm \sqrt{- 160}}{8}$

These roots will be complex numbers

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