Exercise 2.2.3

The quadratic formula:

If \(a x^2 + b x + c = 0\), then \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\).

The expression \(b^2-4ac\) under the radical is called the discriminant.

Some important things to know about the discriminant:

  1. \(a x^2 + b x + c = 0\) has only one solution if and only if the discriminant equals zero.
  2. \(a x^2 + b x + c = 0\) has two real solutions if and only if the discriminant is positive.
  3. \(a x^2 + b x + c = 0\) has two complex solutions if and only if the discriminant is negative.
  4. \(a x^2 + b x + c = 0\) can be factored in the product of two linear factors with integer coefficients if and only if the discriminant is a perfect square.