Question

If a,b are the zeroes of f(x)=
$x {}^{2} + 3x + 1$
and c, d are the zeroes of
$g(x) = x { }^{2} + 4x + 1$
then the value of
$e = (a - c)(b -c )( a+d )(b +d )$
divided by 2
is

Answer 1

${x}^{2} + 3x + 1 = 0. \: so \: a + b = - 3...ab = 1. \\ {x}^{2} + 4x + 1 = 0. \: so \: c + d = - 4...cd = 1. \\ \\ 2e \: = (a - c)(b + d) \times (b - c)(a + d) \\ = (ab + ad - bc - cd) (ab - ac + bd - cd) \\ =(ad - bc) \times (bd - ac) \: \: as \: ab = cd = 1. \\ =ab {d}^{2} - {a}^{2} cd - {b}^{2} cd + ab {c}^{2} \\= {c}^{2} + {d}^{2} - ( {a}^{2} + {b}^{2} ) \\ =(c + d)^{2} - 2cd \: - (a + b)^{2} + 2ab \\ =16 - 2 \times 1 - 9 + 2 = 7 \\ e = \frac{7}{2}$
Answer is e = 7/2.

It's preferable to expand terms and substitute values. We don't need to use calculator.