Question

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`?

Answer 1

Discriminant: `(-16)`
There are no roots (i.e. no solutions in terms of Real numbers)

Explanation:

For the general case: `ax^2+bx+c=0`

The discriminant is `Delta = b^2+4ac`

There are `{("zero roots"," if ",Delta < 0),("one root"," if ",Delta=0),("2 roots"," if ",Delta > 0):}`

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

`x=(-b+-sqrt(Delta))/(2a)`

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For the given example: `x^2-2x+5=0`
`color(white)("XXX")`i.e. `a=1, b=(-2),c=5`

`Delta=(2^2-4*1*5) = -16`

and since `Delta < 0`
`color(white)("XXX")`this equation has no solutions.