Question

Find the discernment and the numbers of the number of real roots for this equation.
x^2+3x+8=0

Answer 1

Answer: 2 distinct complex solutions (ie non real solutions).

Work Shown:

The given equation is in the form ax^2+bx+c = 0, so

a = 1, b = 3, c = 8

Plug those into the formula below to find the discriminant

D = b^2 - 4ac

D = 3^2 - 4(1)(8)

D = -23

The discriminant is negative, so we get two nonreal solutions. The two solutions are complex numbers in the form a+bi, where a & b are real numbers and
`i = √(-1)`. The two solutions are different from one another.

Answer 2

Discriminant: -23

Number of real roots: 0

Explanation:

For a quadratic in standard form
`ax^2+bx+c`, the discriminant is given by
`b^2-4ac`.

In
`x^2+3x+8`, assign:

• 
  `a\implies 1`
• 
  `b\implies 3`
• 
  `c\implies 8`

The discriminant is therefore:

`3^2-4(1)(8)=9-32=\boxed{-23}`

For any quadratic:

• If the discriminant is greater than 0, the quadratic has two real roots
• If the discriminant is equal to 0, the quadratic has one real root
• If the discriminant is less than 0, the quadratic as no real roots

Since the quadratic in the question has a discriminant less than 0, there are no real solutions to this quadratic.