How do you use the discriminant to determine the nature of the roots for #5x^2 - 5x – 60 = 0#?

1 Answer
Jun 15, 2015

Answer:

#5x^2-5x-60 = 5(x^2-x-12)#

#Delta(x^2-x-12) = 7^2#

Since this is positive and a perfect square, the two roots are distinct, real and rational.

Explanation:

#5x^2-5x-60 = 5(x^2-x-12)#

#x^2-x-12# is of the form #ax^2+bx+c# with #a=1#, #b=-1# and #c=-12#.

This has discriminant #Delta# given by the formula:

#Delta = b^2-4ac = (-1)^2 - (4xx1xx-12) = 1+48 = 49 = 7^2#

Since this is positive and a perfect square, #x^2-x-12 = 0#
and hence #5x^2-5x-60=0# has two distinct real, rational roots.

Here are the possible cases:

#Delta > 0# There are two distinct, real roots. If #Delta# is also a perfect square (and the original coefficients are rational), then the roots are also rational.

#Delta = 0# There is one repeated root (with multiplicity 2). If the coefficients of the quadratic are rational, this root is rational too.

#Delta < 0# There are no real roots. There are two distinct complex roots (which are complex conjugates of one another).