Find the nature of the roots of the quadratic equation 3x² - 7x + $\dfrac{1}{2}$ = 0. If real roots exist, find them.

Comparing equation 3x² - 7x + $\dfrac{1}{2}$ = 0, with ax² + bx + c = 0, we get :

a = 3, b = -7 and c = $\dfrac{1}{2}$

D = b² - 4ac

$= (-7)^2 - 4 \times 3 \times \dfrac{1}{2} \\[1em] = 49 - 6 \\[1em] = 43.$

Since, D > 0.

∴ Roots are real and unequal.

By formula,

x = $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting values we get :

$\Rightarrow x = \dfrac{-(-7) \pm \sqrt{43}}{2 \times 3} \\[1em] \Rightarrow x = \dfrac{7 \pm \sqrt{43}}{6} \\[1em]$

Hence, roots of the equation are $\dfrac{7 \pm \sqrt{43}}{6}.$