Using the Discriminant in Factoring
Consider ax^{2} + bx + c, where a, b, and c are integers with a greatest common
factor of 1. If b^{2}  4ac is a perfect square, then
is a whole number,
and the solutions to ax^{2} + bx + c = 0 are rational numbers. If the solutions to a
quadratic equation are rational numbers, then they could be found by the factoring
method. So if b^{2}  4ac is a perfect square, then ax^{2} + bx
+ c factors. It is also
true that if b^{2}  4ac is not a perfect square, then ax^{2} + bx
+ c is prime.
Example 1
Using the discriminant
Use the discriminant to determine whether each polynomial can be factored.
a) 6x^{2} + x  15
b) 5x^{2}  3x + 2
Solution
a) Use a = 6, b = 1, and c = 15 to find b^{2}  4ac:
b^{2}  4ac = 1^{2}  4(6)(15) = 361
Because
, 6x^{2}
+ x  15 can be factored. Using the ac method,
we get
6x^{2} + x  15 = (2x  3)(3x + 5).
b) Use a = 5, b = 3, and c = 2 to find b^{2}  4ac:
b^{2}  4ac = (3)^{2}  4(5)(2) = 31
Because the discriminant is not a perfect square, 5x^{2}  3x + 2 is
prime.
