Question

If '$\alpha$' and '$\beta$' are the roots of quadratic equation $x^2 + (k^2 - 1)x - 20 = 0$ such that $\alpha^2 - \beta^2 - \alpha\beta = 29$ and $\alpha - \beta = 9$, then find the value of $k$.

Accepted Answer

0. Product of roots $\alpha\beta = -20$. Given $\alpha^2 - \beta^2 - \alpha\beta = 29 \implies (\alpha - \beta)(\alpha + \beta) - (-20) = 29$. Since $\alpha - \beta = 9$, $9(\alpha + \beta) + 20 = 29 \implies 9(\alpha + \beta) = 9 \implies \alpha + \beta = 1$. Sum of roots = $-(k^2 - 1) = 1 \implies k^2 - 1 = -1 \implies k^2 = 0 \implies k = 0$.

Value of k in quadratic equation

Explanation

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