Question

if the zeroes of the polynomial ax2+bx+b=0 are in the ratio m:n then find the value of root m/n+rootn/m

Answer 1

this is the solution

Answer 2

$Let \: \alpha \: and \: \beta \: are \: zeroes \\of \: the \: polynomial \: ax^{2}+bx+b = 0$

$\alpha : \beta = m : n\: (given)$

$\implies \frac{\alpha}{\beta} = \frac{m}{n} \: --(1)$

$i )Sum \:of \: the\: zeroes = \frac{-b}{a}\\\implies \alpha + \beta = \frac{-b}{a}\: --(2)$

$ii ) Product \:of \: the\: zeroes = \frac{b}{a}$

$\implies \alpha \beta = \frac{b}{a}\: --(3)$

$Now, \: Value \: of \: \sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} \\= \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} \\=\frac{ \sqrt{\alpha}^{2} + \sqrt{\beta}^{2}}{\sqrt{\alpha \beta}} \\= \frac{\alpha + \beta}{\sqrt{\alpha \beta} }\\= \frac{\frac{-b}{a}}{\sqrt{\frac{b}{a}}} \\= - \sqrt{\frac{b}{a}}$

Therefore.,

$\red {Value \: of \: \sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} } \green {= - \sqrt{\frac{b}{a}} }$

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