Question

if alpha, beta are zeros if x²+5x+1 then find a quadratic polynomial whose zeros are alpha+2, beta+2​

Answer 1

Answer

$\large{\bold{ {x}^{2} + x - 5}}$

Step-by-Step-Explanation:

$If \: \alpha \: and \: \beta \: are \: the \: roots \: \\ of \: A \: Quadratic \: Equation \: \\ then \: the \: equation \: is \: given \: by \: \\ {x}^{2} - ( \alpha + \beta )x + \alpha \beta \\ \\ given \: eqn \: \\ \\ {x}^{2} + 5x + 1 \\ \\ = > \alpha + \beta = - 5 \\ \\ = > \alpha \beta = 1 \\ \\ now \: the \: new \: roots \: are \: \\ \alpha + 2 \: and \: \beta + 2 \\ \\ = > a = \alpha + 2 \\ \\ = > b = \beta + 2 \\ \\ required \: qe \\ \\ {x}^{2} - (a + b)x + ab \\ \\ \\ \\ = > a + b = \alpha + 2 + \beta + 2 \\ \\ = > a + b = (\alpha + \beta ) + 4 \\ \\ = > a + b = - 5 + 4 = - 1 \\ \\ \\ = > ab = ( \alpha + 2)( \beta + 2) \\ \\ = > ab = \alpha \beta + 2( \alpha + \beta ) + 4 \\ \\ = > ab = 1 + 2( - 5) + 4 = - 5 \\ \\ therefore \: the \: eq \: is \\ \\ \\ \\ \bold{ {x}^{2} + x - 5}$

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