How do I find the discriminant?

Answer 1

Given the quadratic equation ax^2 + bx + c =0, where a, b, and c are real numbers:

(The discriminant is equal to b^2 - 4ac)

If b^2 - 4ac < 0, there are two conjugate imaginary roots.

If b^2 - 4ac = 0, there is one real root (called double root)

If b^2 - 4ac > 0, there are two different real roots.

In the special case when the equation has integral coefficients (means that all coefficients are integers), and b^2 - 4ac is the square of an integer, the equation has rational roots. That is , if b^2 - 4ac is the square of an integer, then ax^2 + bx + c has factors with integral coefficients.

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Strictly speaking, the last part of the last sentence is not true.

For example, consider the equation 4x2 + 8x + 3 = 0

the discriminant is 16, which is a perfect square and the equation can be written as (2x+1)*(2x+3) = 0

To that extent the above is correct.

However, the equation can also be written, in factorised form, as (x+1/2)*(x+3/2) = 0 Not all integral coefficients.