Question

If $\alpha ,\beta and\gamma$ are the real roots of the equation $x^3+5x^2+9x-6=0$. Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$.
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• A. 007
• B. seven
• C. Seven
• D. 07
• E. 7.0
• F. 7
• G. 7.00

Answer 1

Answer: (A), (B), (C), (D), (E), (F), (G)

Given: $\alpha ,\beta$ and $\gamma$ are the real roots of the equation $x^3+5x^2+9x-6=0$
Using the relation between roots and coefficients, we can write
$\alpha +\beta +\gamma =-\frac{5}{1}...\left(i\right)\alpha .\beta +\beta .\gamma +\gamma .\alpha =\frac{9}{1}...\left(ii\right)$
Using identity
${(\alpha +\beta +\gamma )}^2={\alpha }^2+{\beta }^2+{\gamma }^2+2(\alpha \beta +\beta \gamma +\gamma \alpha )\Rightarrow {\alpha }^2+{\beta }^2+{\gamma }^2={(\alpha +\beta +\gamma )}^2-2(\alpha \beta +\beta \gamma +\gamma \alpha )$

Using eq.(i) and (ii), we get$\Rightarrow {\alpha }^2+{\beta }^2+{\gamma }^2={(-5)}^2-2\times 9$
= 25 - 18
= 7