Solve the quadratic equation $\dfrac{1}{2}x^2 - \sqrt{11}x + 1 = 0$ by using formula giving the answer correct to three significant figures.

Given,

Equation : $\dfrac{1}{2}x^2 - \sqrt{11}x + 1 = 0$

Comparing above equation with ax² + bx + c = 0, we get :

a = $\dfrac{1}{2},\text{ b} = -\sqrt{11}$ and c = 1.

By formula,

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

Substituting values we get :

$\Rightarrow x = \dfrac{-(-\sqrt{11}) \pm \sqrt{(-\sqrt{11})^2 - 4 \times \dfrac{1}{2} \times 1}}{2 \times \dfrac{1}{2}} \\[1em] \Rightarrow x = \dfrac{\sqrt{11} \pm \sqrt{11 - 2}}{1} \\[1em] \Rightarrow x = \sqrt{11} \pm \sqrt{9} \\[1em] \Rightarrow x = 3.316 \pm 3 \\[1em] \Rightarrow x = 3.316 + 3, 3.316 - 3 \\[1em] \Rightarrow x = 6.316, 0.316$

Since, we need to find value till three significant figures.

Hence, x = 6.31, 0.316