What is the discriminant of #x^2-4=0# and what does that mean?

1 Answer
Jul 16, 2015

Answer:

The discriminant is 8. It tells you that there are two separate real roots to the equation.

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

#x^2 - 2 = 0#

#Δ = b^2 – 4ac = (0)^2 -4×1×(-2) = 0 +8 = 8#

This tells you that there are two separate real roots.

We can see this if we solve the equation.

#x^2 -2 = 0#

#x = (-b±sqrt(b^2-4ac))/(2a) = (-0±sqrt((0)^2 -4×1×(-2)))/(2×1) = ±sqrt(0+8)/2 = ±sqrt8/2 = ±(2sqrt2)/2 = ±sqrt2##

#x = sqrt2# and #x = -sqrt2#

There are two separate real roots to the equation.