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

1 Answer
Stefan V.
Jul 30, 2015

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

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