If the roots of the quadratic equation x^2 + px + q = 0 are [ 30^ ] and [ 15^ ], respectiv…

Question

If the roots of the quadratic equation ${x^2} + px + q = 0$ are $$\tan {30^ \circ }$$ and $$\tan {15^ \circ }$$, respectively, then the value of $2 + q - p$  is

Accepted Answer

${x^2} + px + q = 0$

Sum of roots $$ = \tan {30^ \circ } + \tan {15^ \circ } =  - p$$

Products of roots $$ = \tan {30^ \circ }.\tan {15^ \circ } = q$$

$$\tan {45^ \circ } = {{\tan {{30}^ \circ } + \tan {{15}^ \circ }} \over {1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {{ - p} \over {1 - q}} = 1$$

$$ \Rightarrow  - p = 1 - q \Rightarrow q - p = 1$$

$$\therefore$$ $2 + q - p = 3$

Solutions · 1