Question

if alpha and beta are in the roots of the equation 3x2-6x+4=0 find the value of alpha^2 and beta^2​

Answer 1

/* There is a mistake in the question. It may be like this . */

$Given \: \alpha \: and \: \beta \: are \: roots \\of \: a \: Quadratic \: equation \: 3x^{2} - 6x + 4 = 0$

$Compare \:above \: equation \: with \\ax^{2} + bx + c = 0 \:, we \:get$

$a = 3 , b = -6 \: and \: c = 4$

$i ) Sum \:of \: the \: roots = \frac{-b}{a}$

$\implies \alpha + \beta = \frac{ -(-6)}{3}$

$\implies \alpha + \beta = \frac{ 6}{3}$

$\implies \alpha + \beta = 2 \: --(1)$

$ii ) Product \:of \: the \: roots = \frac{c}{a}$

$\implies \alpha \beta = \frac{ 4}{3}\: --(2)$

$Now, \red { \alpha^{2} + \beta^{2} } \\= ( \alpha + \beta )^{2} - 2 \alpha \beta \\= 2^{2} - 2 \times \frac{ 4}{3} \\= 4 - \frac{ 8}{3} \\= \frac{12 - 8 }{3} \\= \frac{ 4}{3}$

Therefore.,

$\red {Value \:of \: \alpha^{2} + \beta^{2} }\green { = \frac{ 4}{3}}$

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