Using the Discriminant in Factoring
Consider ax2 + bx + c, where a, b, and c are integers with a greatest common
factor of 1. If b2 - 4ac is a perfect square, then
is a whole number,
and the solutions to ax2 + 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 b2 - 4ac is a perfect square, then ax2 + bx
+ c factors. It is also
true that if b2 - 4ac is not a perfect square, then ax2 + bx
+ c is prime.
Using the discriminant
Use the discriminant to determine whether each polynomial can be factored.
a) 6x2 + x - 15
b) 5x2 - 3x + 2
a) Use a = 6, b = 1, and c = -15 to find b2 - 4ac:
b2 - 4ac = 12 - 4(6)(-15) = 361
+ x - 15 can be factored. Using the ac method,
6x2 + x - 15 = (2x - 3)(3x + 5).
b) Use a = 5, b = -3, and c = 2 to find b2 - 4ac:
b2 - 4ac = (-3)2 - 4(5)(2) = -31
Because the discriminant is not a perfect square, 5x2 - 3x + 2 is