Question

findthe zeroes of the quadratic polynomial 7y^2-11/3y-2/3and verify the relationship between the cofficeient​

Answer 1

$\large\underline{\bf{Solution-}}$

$\bf :\longmapsto\:Let \: f(y) = {7y}^{2} - \dfrac{11}{3} y - \dfrac{2}{3}$

$\rm \:= \: \:\dfrac{ {21y}^{2} - 11y - 2 }{3}$

$\rm \:= \: \:\dfrac{1}{3}\bigg( {21y}^{2} - 14y + 3y - 2 \bigg)$

$\rm \:= \: \:\dfrac{1}{3}\bigg(7y(3y - 2) + 1(3y - 2)\bigg)$

$\rm \:= \: \:\dfrac{1}{3}(7y + 1)(3y - 2)$

Hence,

$\rm :\longmapsto\:Zeroes \: of \: f(y) \: = - \dfrac{1}{7} \: \: or \: \: \dfrac{2}{3}$

$\rm :\longmapsto\:Let \: \alpha = - \dfrac{1}{7} \: \: and \: \: \beta = \dfrac{2}{3}$

Consider,

$\rm :\longmapsto\: \alpha + \beta$

$\rm \:= \: \: - \dfrac{1}{7} + \dfrac{2}{3}$

$\rm \:= \: \: \dfrac{ - 3 + 14}{21}$

$\rm \:= \: \: \dfrac{ 11}{21}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \bf\implies \:\bf \: \alpha + \beta = \dfrac{11}{21}}$

Consider,

$\rm :\longmapsto\: \alpha \beta$

$\rm \:= \: \: - \dfrac{1}{7} \times \dfrac{2}{3}$

$\rm \:= \: \: - \dfrac{2}{21}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \bf\implies \:\bf \: \alpha \beta = - \: \dfrac{2}{21}}$

Now, Verification

On comparing the given polynomial with

$\rm :\longmapsto\: {ay}^{2} + by + c$

We get

$\red{\rm :\longmapsto\:a = \: 7 \: \: \: \: \: \: } \\ \red{\rm :\longmapsto\: b = \: - \dfrac{11}{3} } \\ \red{\rm :\longmapsto\: c = \: - \dfrac{2}{3} \: \: }$

We know that

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ y}{coefficient\ of\ y^{2}}}}$

$\rm :\implies\:Sum \: of\ the\ zeroes = - \dfrac{ - \dfrac{11}{3} }{ \: \: 7 \: \: } = \dfrac{11}{21}$

And

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ y^{2}}}}$

$\rm :\implies\:{{\sf Product\ of\ the\ zeroes=\dfrac{\dfrac{ - 2}{3} }{7}}} = - \: \dfrac{2}{21}$

Hence,

We conclude that

$\boxed{\red{\rm :\longmapsto\:\sf Sum\ of\ the\ zeroes= \alpha + \beta }}$

and

$\boxed{\red{\rm :\longmapsto\:\sf Product\ of\ the\ zeroes= \alpha \beta }}$

Hence, Verified