Question

If one zero of the polynomial 5x square + 13x - a is the reciprocal of the other. find a

Answer 1

let the zeroes be a and 1/a

$5x {}^{2} + 13x - a$

let zeroes be,

$a \: and \: \frac{1}{a}$

product of zeroes =

$\frac{c}{a}$

so,

$a + \frac{1}{a } = \frac{ - a}{5} \\ \frac{ {a}^{2} + 1}{a} = \frac{ - a}{5} \\ 5 {a}^{2} + 5 = - { a}^{2} \\ 6 {a}^{2} + 5 = 0 \\ {a}^{2} = \frac{ - 5}{6} \\ a = \sqrt{ \frac{ - 5}{6} }$

as there is negative sign in square root

hence no real roots exist for a