Question

If alfa and bita are zeroes of the polynomial f (x)=x2-x-k find that alfa-bita=9 find k

Answer 1

$Answer \: \\ \\ GIVEN \: \: QUESTION \: \: Is \: \: \\ \\ f(x) = x {}^{2} - x + k \\ \alpha - \beta = 9 \: \: \: \: \: \: \: \: \: find \: k \: \\ \\ f(x) = x {}^{2} - x + k \\ \\ \alpha \: + \beta = \frac{1}{1} \: \: \: \: and \: \: \: \alpha \beta = \frac{k}{1} \\ \\ \alpha - \beta = \sqrt{( \alpha + \beta ) {}^{2} - 4 \alpha \beta } \\ \\ \sqrt{1 {}^{2} - 4k } = 9 \\ squaring \: \: on \: both \: sides \: we \: have \\ \\ ( \sqrt{1 - 4k} ) {}^{2} = 9 {}^{2} \\ \\ 1 - 4k = 81 \\ \\ 1 - 81 = 4k \\ \\ - 80 = 4k \\ \\ k = \frac{ - 80}{4} \\ \\ k = - 20 \\ \\ \\ Note \: \: \: \: \: \\ \\1) \: \: \: a - b = \sqrt{(a + b) {}^{2} - 4ab } \\ \\ 2) \: \: \: for \: a \: general \: \: quadratic \: polynomial \: say \\ p(x) = ax {}^{2} + bx + c \\ \\ \alpha + \beta = \frac{ - b}{a} \: \: \: \: and \: \: \: \: \alpha \beta = \frac{c}{a}$