Question

The quadratic equation with rational coefficients having $\frac{\sqrt{3}}{2}$ as a root is _______.

Answer 1

Irrational roots of a quadratic equation always occurs in pairs.

If one root is $\frac{\sqrt{3}}{2}$ then, other root is –$\frac{\sqrt{3}}{2}$.

Now,

Quadratic equation with given roots is
x² – (sum of roots)x + (product of roots) = 0

$$\begin{gathered} \Rightarrow x^2-\left(\frac{\sqrt{3}}{2}+\left(-\frac{\sqrt{3}}{2}\right)\right)x+\left(\frac{\sqrt{3}}{2}\times -\frac{\sqrt{3}}{2}\right)=0 \\ \Rightarrow x^2+\left(\frac{-3}{4}\right)=0 \\ \Rightarrow x^2-\frac{3}{4}=0 \end{gathered}$$

Hence, quadratic equation with rational coefficients having $\frac{\sqrt{3}}{2}$ as a root is $\underline{x^2-\frac{3}{4}=0}.$