Question

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.

Answer 1

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.