Question

If \( \tan 15^\circ \) and \( \tan 30^\circ \) are the roots of the equation \( x^2 + px + q = 0 \), then \( pq = \):

• A. \( \frac{6\sqrt{3} + 10}{\sqrt{3}} \)
• B. \( \frac{10 - 6\sqrt{3}}{3} \)
• C. \( \frac{10 + 6\sqrt{3}}{3} \)
• D. \( \frac{10 - 6\sqrt{3}}{3} \)

Hint:

The sum and product of roots for a quadratic equation \( x^2 + px + q = 0 \) are given by: \[ {Sum of roots} = -p \] \[ {Product of roots} = q \]

Answer 1

The Correct Option is B

Step 1: {Apply Vieta's formulas for roots}
For a quadratic equation of the form \( x^2 + px + q = 0 \):\[p = -(\tan 15^\circ + \tan 30^\circ)\]\[q = \tan 15^\circ \tan 30^\circ\]Step 2: {Determine the values of \( p \) and \( q \)}
\[\tan 15^\circ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\]\[\tan 30^\circ = \frac{1}{\sqrt{3}}\]Step 3: {Calculate the product \( pq \)}
\[pq = -\frac{4(\sqrt{5} - 1)}{3(\sqrt{5} + 1)^2}\]Step 4: {Simplify the expression}
\[pq = \frac{10 - 6\sqrt{3}}{3}\]