Question

If 2+√3 is one zero of x^2+px+q find p and q

Answer 1

put the value in eq
${x}^{2} + px + q \\ ( {2 + \sqrt{3}) }^{2} + p(2 + \sqrt{3} ) + q = 0 \\ 4 + 3 + 4 \sqrt{3} + 2p + \sqrt{3}p + q = 0 \\ 2p + \sqrt{3} p + q = - 7 - 4 \sqrt{3} \: \: \: \: \: eq1 \\ we \: \: know \: that \: complex \\ \: zeros \: occurs \: in \: pairs \: \: so \: other \: \\ zero \: will \: be \: 2 - \sqrt{3} \\ 4 + 3 - 4 \sqrt{3} + 2p - \sqrt{3}p + q = 0 \\ 2p - \sqrt{3}p + q = - 7 + 4 \sqrt{3} \: \: \: eq2 \\subtract \: both \: equations \\ 2 \sqrt{3} p = - 8 \sqrt{3} \\ p = \frac{ - 8 \sqrt{3} }{2 \sqrt{3} } = - 4 \\ \\ add \: both \: equation \\ 4p + 2q = - 14 \\ 2p + q = - 7 \\ - 8 + q = - 7 \\ q = - 7 + 8 \\ q = 1 \\ so \: \: \: p = - 4 \: \: \: \: q = 1 \: \: \: \: ans$