Question

What is the discriminant of `5x^2 + 11x + 8 = 0` and what does that mean?

Answer 1

The discriminant is -39. It tells you that there are no real roots to the equation, but there are two complex roots.

Explanation:

If you have a quadratic equation of the form

`ax^2+bx+c=0`

The solution is

`x = (-b±sqrt(b^2-4ac))/(2a)`

The discriminant `Δ` is `b^2 -4ac`.

The discriminant "discriminates" the nature of the roots.

There are three possibilities.

• If `Δ > 0`, there are two separate real roots.
• If `Δ = 0`, there are two identical real roots.
• If `Δ <0`, there are no real roots, but there are two complex roots.

Your equation is

`5x^2 +11x +8 = 0`

`Δ = b^2 – 4ac = 11^2 -4×5×8 = 121 – 160 = -39`

This tells you that there are no real roots, but there are two complex roots.

We can see this if we solve the equation.

`5x^2 +11x +8 = 0`

`x = (-b±sqrt(b^2-4ac))/(2a) = (-11±sqrt(11^2 -4×5×8))/(2 ×5) = (-11±sqrt(121-160))/10 = (-11±sqrt(-39))/10 = 1/10(-11 ±isqrt39) = -1/10(11±isqrt39)`

`x = -1/10(11+isqrt39)` and `x = -1/10(11-isqrt39)`

There are no real roots, but there are two complex roots to the equation.