Using the discriminant, give the nature of the roots of #7x^3+x^2-35x=5#?

Using the discriminant, give the nature of the roots of #7x^3+x^2-35x=5#.
Also solve the equation.

1 Answer
Mar 13, 2017

Answer:

We have real roots.

Explanation:

The discriminant of a cubic equation #ax^3+bx^2+cx+d=0# is #b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#.

  • If discriminant is positive, we have real roots;
  • if it is zero it has repeated roots and all real; and
  • if it is less than zero, we have one rel root and two complex conjugates.

In the given equation #7x^3+x^2-35x-5=0#, discriminant is

#1^2xx(-35)^2-4xx7xx(-35)^3-4xx1^3xx(-5)-27xx7^2xx(-5)^2+18xx7xx1xx(-35)xx(-5)#

= #1225+1200500+20- 33075+22050=1190720#

Hence, we have real roots.