Question

If α and β are the zeroes of the polynomial 4x² -2x + ( k -4) and α=1/β ,find the value of k.

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Answer 1

Given :-

• α and β are the zeroes of the polynomial 4x² -2x + ( k -4) and α=1/β

To Find :-

• Value of k.

Solution :-

We know that,

$\sf \: If \: \alpha \:and \: \beta \: are \: zeroes \: of \: {ax}^{2} + bx + c \: then$

$\boxed{ \bf \: Product \: of \: zeroes = \: \dfrac{constant \: term}{coefficient \: of \: {x}^{2}}}$

Given that,

$\sf \: \alpha \: and \: \beta \: are \: zeroes \: of \: {4x}^{2} - 2x + (k - 4)$

So,

$\rm :\longmapsto\: \alpha \beta = \dfrac{k - 4}{4}$

But,

$\rm :\longmapsto\: \alpha = \dfrac{1}{ \beta }$

$\rm :\implies\:\: \dfrac{1}{ \beta } \times \beta = \dfrac{k - 4}{4}$

$\rm :\longmapsto\:\dfrac{k - 4}{4} = 1$

$\rm :\longmapsto\:k - 4 = 4$

$\rm :\longmapsto\:k = 4 + 4$

$\rm :\longmapsto\:k = 8$

Additional Information :-

$\sf \: If \: \alpha \:and \: \beta \: are \: zeroes \: of \: {ax}^{2} + bx + c \: then$

$\boxed{ \bf \: Sum \: of \: zeroes = - \: \dfrac{coefficient \: of \: {x}}{coefficient \: of \: {x}^{2}}}$

$\rm :\longmapsto\: { \alpha }^{2} + { \beta }^{2} = {( \alpha + \beta )}^{2} - 2 \alpha \beta$

$\rm :\longmapsto\: { \alpha }^{3} + { \beta }^{3} = {( \alpha + \beta )}^{3} - 3\alpha \beta ( \alpha + \beta )$

$\rm :\longmapsto\: {( \alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta$