Question

If roots α, β of the equation $x^2-px+16=0$ satisfy the relation α² + β² = 9, then write the value p.

Answer 1

Given equation: $x^2-px+16=0$
Also, $\alpha$ and $\beta$ are the roots of the equation satisfying ${\alpha }^2+{\beta }^2=9.$
From the equation, we have:
Sum of the roots = $\alpha +\beta =$ $-\left(\frac{-p}{1}\right)=p$

Product of the roots = $\alpha \beta$ =$\frac{16}{1}=16$


$$\begin{gathered} \mathrm{Now,}{\left(\alpha +\beta \right)}^2={\alpha }^2+{\beta }^2+2\alpha \beta \\ \Rightarrow p^2=9+32 \\ \Rightarrow p^2=41 \\ \Rightarrow p=\sqrt{41} \end{gathered}$$

Hence, the value of $pis\sqrt{41}.$