Math, asked by Amirdhavarshini3142, 9 months ago

If one zero of the polynomial (a2 + 9)x2
+ 13x + 6a is reciprocal of the other.
Find "a2 + 2a + 5" .

Answers

Answered by amansharma264
3

EXPLANATION.

One zero of the polynomial =>

( a² + 9 )x² + 13x + 6a is reciprocal of the other.

Find a² + 2a + 5.

 \sf : \implies \: one \: zeroes \: of \: the \: polynomial \: ( {a}^{2} + 9) {x}^{2} + 13x + 6a \\  \\ \sf : \implies \: products \: of \: roots \: of \: the \: equation \:  \alpha  \beta  =  \frac{c}{a}  =  \frac{6a}{ {a}^{2} + 9 }  \\  \\ \sf : \implies \:  \: one \: roots \: is \: reciprocal \: to \: other \:  \:  =  \alpha  \times  \frac{1}{ \alpha } = 1

 \sf : \implies \:  \dfrac{6a}{ {a}^{2}  + 9} = 1 \\  \\  \sf : \implies \: 6a = 9 +  {a}^{2} \\  \\  \sf : \implies \:  {a}^{2} - 6a  - 9 = 0 \\  \\  \sf : \implies \:  {a}^{2}  - 3a - 3a - 9 = 0 \\  \\ \sf : \implies \: a(a - 3) - 3(a - 3) = 0 \\  \\ \sf : \implies \: (a - 3)(a - 3) = 0

\sf : \implies \: (a - 3) {}^{2}  = 0 \\  \\ \sf : \implies \: a = 3 \:

 \sf : \implies \: equation \:  =  {a}^{2}  + 2a + 5 = 0 \\  \\ \sf : \implies \: (3) {}^{2}  + 2(3) + 5 \\  \\ \sf : \implies \: 9 + 6 + 5 = 20

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