Question

if alpha beta are roots of the equation X square + 5 x + 5 = 0 then equation whose roots are alpha plus one and pita plus one is​

Answer 1

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$\underline{\mathrm{The\: quadratic\: equation\: whose\: roots\: are\: (\alpha\: +\: 1)\:(\beta \: +\: 1)\: is\: x^2\:+\: 3x\: +\: 1\: = \:0}}$

$\mathrm{Given\: equation\: is}$

• $x^2\:+\:5x\:+\:5\:=\:0\:\:\:\:\:(1)$    

$\mathrm{Comparing\: with\:ax^2\:+\:bx\:+\:c ,\: we\: get}$

• $\mathrm{a\: =\: 1,\: b\: =\: 5,\: c\: =\: 5}$

$\mathrm{Roots\: of\: (1)\: are\:\alpha ,\: \beta}$

• $\mathrm{Sum\:of\:roots\:=\:\alpha \:+\:\beta \:=\:\frac{-b}{a}\:=\:\frac{-5}{1}\:=\:-5\:\:\:\:\:\:\:\:(2)}$  
• $\mathrm{Product\: of\: roots\: =\: \alpha\beta\: =\:\frac{c}{a}\:=\:\frac{5}{1}\: =\:5\:\:\:\:\:\:\:(3)}$

$\mathrm{Quadratic\: equation\: with\: roots\:( \alpha \: +\: 1)\:(\beta \: +\: 1 )\:is\: given\: by}$

$\sf{x^2\: +\:(sum\: of\: roots)x\: +\: (product\: of\: roots)\: =\: 0}$

$\sf{x^2\:+\:(\alpha\:+\:1\: +\:\beta\:+\: 1)x\: +\: (\alpha +1)(\beta +1)\: =\: 0}$

$\sf{x^2\:+\:(\alpha \: +\: \beta \: +\: 2)x\: +\: \alpha\beta \: +\: \alpha \: +\:\beta \: +\: 1\: =\: 0}$

$\sf{x^2\:+\:(-5\:+\:2)x\: +\: 5\: +\: (-5)\: +\: 1\: =\: 0}$      

$\mathrm{[from\: (2)\: and\: (3)]}$

$\sf{x^2\:+\:(-3)x\: +\: 5\: -\: 5\: +\: 1\: =\: 0}$

$\sf{x^2\:+\: 3x\: +\: 1\: =\: 0}$

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Answer 2

Answer:

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