Question

How do you describe the nature of the roots of the equation `7x^2=4x+1`?

Answer 1

Roots are irrational. In the given equation these are
`-2/7-sqrt11/7` and
`-2/7+sqrt11/7`

Explanation:

Nature of roots of an equation `ax^2+bx+c=0`/ are determined by its discriminant, which is `b^2-4ac`.

Assuming that the coefficients `a`, `b` and `c` are rational,

if discriminant is zero, there is one root and we also say roots coincide.

if discriminant is negative, roots are complex,

if discriminant is positive and square of a rational number, roots are rational

and if discriminant is positive but not a square of a rational number, roots are irrational.

As the equation `7x^2=4x+1`
`hArr7x^2-4x-1=0`, the discriminant is `(-4)^2-4×7×(-1)=16+28=44`, which is positive but not a square of a rational number. Hence roots are irrational

In fact, according to quadratic formula roots are

`((-4)+-sqrt44)/(2×7)` or

`-4/14+-2sqrt11/14` or

`-2/7-sqrt11/7` and
`-2/7+sqrt11/7`