Question

If one zero of the polynomial (a^+9)x^+13x+6a is a reciprocal of the other , fimd the value of a

Answer 1

let one root is alpha and another is 1/ alpha
$\alpha + \frac{1}{ \alpha } = \frac{ - b}{a} = \frac{ - 13 }{ {a}^{2} + 9} \\ \alpha \times \frac{1}{ \alpha } = \frac{c}{a} = \frac{6a}{ {a}^{2} + 9}$
from Multiplication of roots,we get
$\frac{6a}{ {a}^{2} + 9 } = 1 \\ 6a = {a}^{2} + 9 \\ {a}^{2} - 6a + 9 = 0$
factorise the equation to get the value of a
${a}^{2} - 3a - 3a + 9 = 0 \\ a(a - 3) - 3(a - 3) = 0 \\ (a - 3)(a - 3) = 0 \\ a = 3$
so the value if a = 3 from both the factors.
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