Question

if zeros alpha and beta of a polynomial x²-7z+k are such that alpha-beta=1, find the value of k

Answer 1

$\sf \bf Given: \sf The\ zeroes\ are\ \alpha\ and\ \beta\ and\ that\ \alpha\ +\ \beta\ =\ 1.  \\\\ \bf To\ find: \sf The\ value\ of\ k.\\\\\bf Answer: \\\\\sf We\ know\ that\ \alpha\ +\ \beta\ =\ 1.\\\\\sf We\ know\ that\ the\ sum\ of\ the\ zeroes\ of\ a\ polynomial\ =\ \frac{-b}a}\\\\\sf x^{2} - 7z + k\\\\From\ the\ above\ equation,\ the\ sum\ of\ the\ zeroes\ =\ \frac{-7}{1}.$$\sf Taking\ the\ sum\ in\ terms\ of\ \alpha\ and\ \beta,\\\\\alpha\ +\ \beta\ =\ -7\ (Taking\ this\ as\ equation\ 1.)\\\\Let\ \alpha\ -\ \beta\ =\ 1\ be\ equation\ 2.\\\\From\ equation\ 1,\\\\\beta\ =\ -7\ -\ \alpha\\\\Substituting\ the\ above\ value\ in equation\ 2,\\\\\alpha\ -\ (-7\ -\ \alpha)\ =\ 1\\\\\alpha\ +\ 7\ +\ \alpha\ =\ 1\\\\2\alpha\ =\ 1\ -\ 7.\\\\\alpha\ =\ \frac{-6}{2}\ =\ -3\\\\Using\ the\ above\ value\ to\ find\ \beta,\\\\-3\ -\ \beta\ =\ 1\\\\-\beta\ =\ 4$$\\\sf \beta\ =\ -4\\\\Now,\ since\ we\ know\ the\ values\ of\ \alpha\ and\ \beta,\ the\ value\ of\ k\ can\ be\ found.\\\\We\ know\ that\ the\ product\ of\ the\ zeroes\ =\ \frac{c}{a}\\ \\Multiplying\ the\ zeroes,\\\\-3\ \times\ -4\ =\ 12\\\\Hence,\ k\ =\ 12.$