Question

if alpha and beta are the zeros of the quadratic polynomial X square + 5 x minus 5 then (a) alpha + beta is equal to Alpha Beta (b) alpha-beta = alpha beta ​

Answer 1

$Compare \:x^{2} + 5x - 5 \:with\:ax^{2}+bx+c ,\\we \:get$

$a = 1 , b = 5 \:and \: -5$

$\alpha \:and \:\beta \:are \: two \: zeroes \:of \\given \: polynomial$

$i) sum \:of \: the \: zeroes = \frac{-b}{a}$

$\implies \alpha + \beta = \frac{-5}{1}$

$\implies \alpha + \beta = -5\: --(1)$

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

$\implies \alpha \beta = \frac{-5}{1}$

$\implies \alpha \beta = -5\: --(2)$

/* From (1) and (2), we conclude that */

$a) \green {\alpha + \beta = \alpha \beta}$

$b) ( \alpha - \beta)^{2} \\= ( \alpha + \beta)^{2} - 4\alpha \beta \\= (-5)^{2} - 4 \times (-5) \\= 25 + 20 \\= 45$

$Now, \red{ (\alpha - \beta) } = \sqrt{45} \\= \sqrt{3^{2} \times 5 } \\\green { = 3\sqrt{5}}$

Therefore.,

$a) \green {\alpha + \beta = \alpha \beta}$

$b) \red{ (\alpha - \beta) }\green {=3\sqrt{5}} \pink {\neq}{ \alpha \beta }$

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