Question

alpha and bita are two zero of x^-3x+2 find the quadratic equation which zero is alpha+2and bita+2​

Answer 1

➫Question :-

$\rm if \: \alpha \: and \: \beta \: are \: two \: zero\: of \: {x}^{2} - 3x + 2 \\ \rm find \: the \: quadratic \: equation \: whose \: \\ \rm zeros \: are \: \alpha + 2 \: and \: \beta \: + 2.$

➫Answer :-

$\boxed{ \to \rm {x}^{2} - 5x + 12 = 0} \:$

➫Solution :-

According to the question,

$\rm \alpha \: and \: \beta \: are \: two \: zero\: of \: {x}^{2} - 3x + 2 \: \\ \sf therefore \\ \\ \star \rm sum \: of \: zeros \: = \frac{ - coefficient \: of \: x}{coefficient \: of \: {x}^{2} } \\ \\ \mapsto \: \alpha + \beta = \frac{ - ( - 3)}{1} \\ \\ \mapsto \: \alpha + \beta = 3 \: \: .......eq.(1) \\ \\ \sf and \: \\ \\ \star \rm product \: of \: zeros = \frac{ constant \: term}{coefficient \: of \: {x}^{2} } \\ \\ \to \: \alpha \beta = \frac{2}{1} \\ \\ \to \alpha \beta = 2 \: \: .......eq.(2) \\ \\ now \\ \rm we \: need \: a \: quadratic \: whose \: zeros \: are \\ \\ \to \alpha + 2 \: \: and \: \beta + 2 \\ \\ \star \rm sum \: of \: zeros \: of \:required \: quadratic \: is \\ \\ \mapsto \: \alpha + 2 + \beta + 2 \\ \\ \mapsto \boxed{ \rm sum \: of \: zeros = \alpha + \beta + 4} \\ \\ \star \rm \: product \: of \: required \: quadratic \: is \: \\ \\ \mapsto \: ( \alpha + 2)( \beta + 2) \\ \\ \mapsto \: \alpha \beta + 2 \alpha + 2 \beta + 4 \\ \\ \mapsto \: \alpha \beta \: + 2( \alpha \: + \beta) + 4 \\ \\ \boxed{ \rm product \: of \: zeros = \alpha \beta + 2( \alpha + \beta) + 4} \\ \\ \sf \: hence \\ \\ \sf from \: eq.(1) \\ \\ \rm sum \: of \: zeros \:of \: required \: quadratic \\ \: \: \: = 3 + 2 = 5 \\ \\ \bf and \: \\ \\ \rm product \: of \: roots \: of \: required \: quadratic \: \\ \: \: = 2 + 2( 3) + 4 = 12 \\ \\ \sf \: now \: we \: know \: that \: \\ \sf required \: quadratic \: equation \: is \: \\ \\ \to \rm {x}^{2} - (sum \: of \: zeros)x + product \: of \: zeros = 0 \\ \\ \boxed{ \to \rm {x}^{2} - 5x + 12 = 0} \\ \\ \rm this \: is \: the \: required \: quadratic \: equation \: \\ \rm whose \: sum \: of \: zeros \: is \: 5 \: and \: product \: of \: \\ \rm zeros \: is \: 12$