What is the discriminant of #x^2-9=0# and what does that mean?

1 Answer
Jul 30, 2015

Answer:

In your case, #Delta = 36#, which means that your equation has two distinct, rational solutions.

Explanation:

The general form of a quadratic equation is

#ax^2 + bx + c = 0#

for which the discriminant is defined as

#Delta = b^2 - 4 * a * c#

In your case, #a=1#, #b=0#, and #c=-9#, so the discriminant becomes

#Delta = 0^2 - 4 * 1 * (-9) = color(green)(36)#

A quadratic equation that has #Delta>0# has two distinct, real solutions. Moreover, since #Delta# is a perfect square, two two solutions will be rational numbers.

The general form for the solutions of a quadratic equation is

#color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)#

In your case, those two solutions will be

#x_(1,2) = (0 +- 6)/2 = {(x_1 = 3), (x_2 = -3) :}#

Note that these solutions could have easily been determined by

#x^2 - color(red)(cancel(color(black)(9))) + color(red)(cancel(color(black)(9))) = 9#

#sqrt(x^2) = sqrt(9) => x_(1,2) = +-3#