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

Question

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

Aicharya · Accepted Answer

$x^2+p x+q=0$ $	an 30^{\circ}+	an 15^{\circ}=-p$ $	an 30^{\circ} \cdot 	an 15^{\circ}=q$ $	an 45^{\circ}=\frac{	an 30^{\circ}+	an 15^{\circ}}{1-	an 30^{\circ} 	an 15^{\circ}}=\frac{-p}{1-q}=1$ $\Rightarrow-p=1-q$ $\Rightarrow q-p=1 \quad 	herefore 2+q-p=3$.

If the roots of the quadratic equation $x^2+p x+q=0$ are $\tan 30^{\circ}$ and $\tan 15^{\circ}$,…

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