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

Aug 31, 2016

Roots are irrational. In the given equation these are
$- \frac{2}{7} - \frac{\sqrt{11}}{7}$ and
$- \frac{2}{7} + \frac{\sqrt{11}}{7}$

#### Explanation:

Nature of roots of an equation $a {x}^{2} + b x + c = 0$/ are determined by its discriminant, which is ${b}^{2} - 4 a c$.

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 $7 {x}^{2} = 4 x + 1$
$\Leftrightarrow 7 {x}^{2} - 4 x - 1 = 0$, 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

$- \frac{4}{14} \pm 2 \frac{\sqrt{11}}{14}$ or

$- \frac{2}{7} - \frac{\sqrt{11}}{7}$ and
$- \frac{2}{7} + \frac{\sqrt{11}}{7}$