Question

If $\alpha +\beta =8$ and ${\alpha }^2+{\beta }^2=34$, find the quadratic equation whose roots are $\alpha ,\beta$.

Answer 1

$\begin{array}{c}Given,\\ \alpha +\beta =\mathrm{8.................}\left(1\right)\\ {\alpha }^2+{\beta }^2=\mathrm{34...............}\left(2\right)\end{array}$

Squaring equation $\left(1\right)$

$\begin{array}{c}{\left(\alpha +\beta \right)}^2=64\\ {\alpha }^2+{\beta }^2+2\alpha \beta =64\end{array}$

Using equation $\left(2\right)$

$\begin{array}{c}2\alpha \beta \\ =64-34\\ =30\\ \Rightarrow \alpha \beta =15\end{array}$

Equation whose roots are $\alpha$ and $\beta$ is represented as:

$\begin{array}{c}\\ x^2-\left(\alpha +\beta \right)x+p=0\\ x^2-8x+15=0\end{array}$