Question

If aplha and Beta are the zeroes of quadratic polynomial 3x2 + 2x - 7, then the value of (alpha+ 3)(beta+ 3) is ​

Answer 1

$given \: quadratic \: polynomial \: p(x) = 3 {x}^{2} + 2x - 7 \\ \\ on \: comparing \: with \: a {x}^{2} + bx + c \\ \\ \: we \: have \: a = 3,b = 2,c = - 7 \\ \\ if \: \alpha ,\beta \: are \: zeroes \: ,then \: \\ \\ sum \: of \: the \: zeroes \: ( \alpha + \beta ) = \frac{ - b}{a} \\ :• \alpha + \beta = \frac{ - 2}{3} \\ \\ product \: of \: zeroes( \alpha \beta ) = \frac{c}{a } \\ :• \alpha \beta = \frac{ - 7}{3} \\ \\ now( \alpha - 3)( \beta - 3) = \alpha \beta - 3 \beta - 3 \alpha + 9 \\ \\ = \alpha \beta - 3( \alpha + \beta ) + 9 \\ \\ = \frac{ - 7}{3} - 3( \frac{ - 2}{3} ) + 9 \\ \\we \: cut \: three \: number \: because \: it \: is \: common \\ \: in \: numerator \: and \: denominator. \: \\ \\ = \frac{ - 7}{3} - ( - 2) + 9 \\ \\ = \frac{ - 7}{3} + 2 + 9 \\ \\ = \frac{ - 7}{3} + 11 \\ \\ = \frac{ - 7 + 33}{3} \\ \\ :• ( \alpha - 3)( \beta - 3) = \frac{26}{3} \: \: \: ans.$