Question

How do you use the discriminant to determine the nature of the roots for `x^2 + 2x + 5 = 0`?

Answer 1

As `color(red)(Delta = -16`(less than zero), this equation has two complex roots.

Explanation:

`x^2 + 2x + 5 = 0`

The equation is of the form `color(blue)(ax^2+bx+c=0` where:
`a=1, b=2, c=5`

The Discriminant is given by:
`Delta=b^2-4*a*c`
`= (2)^2-(4*(1)*5)`
`= 4-20=-16`

When, `Delta<0` there are two complex solutions.
Here, `color(red)(Delta = -16)`, so this equation has two complex roots

• Note :
  The solutions are normally found using the formula
  `x=(-b+-sqrtDelta)/(2*a)`
  Finding the solutions:
  `x =( -2+-sqrt-16)/(2a`
  `color(red)( x =( -2+4i)/2` and `color(red)(x = (-2-4i)/2`