# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given x^2-2x+5=0?

Dec 17, 2016

Discriminant: $\left(- 16\right)$
There are no roots (i.e. no solutions in terms of Real numbers)

#### Explanation:

For the general case: $a {x}^{2} + b x + c = 0$

The discriminant is $\Delta = {b}^{2} + 4 a c$

There are $\left\{\begin{matrix}\text{zero roots" & " if " & Delta < 0 \\ "one root" & " if " & Delta=0 \\ "2 roots" & " if } & \Delta > 0\end{matrix}\right.$

The solution(s) if it/they exist are given by the quadratic formula:

$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

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For the given example: ${x}^{2} - 2 x + 5 = 0$
$\textcolor{w h i t e}{\text{XXX}}$i.e. $a = 1 , b = \left(- 2\right) , c = 5$

$\Delta = \left({2}^{2} - 4 \cdot 1 \cdot 5\right) = - 16$

and since $\Delta < 0$
$\textcolor{w h i t e}{\text{XXX}}$this equation has no solutions.