Question

If @ and B are the zeroes of quadratic polynomial x^2-kx+15 such that (@^2 + B^2) -2@B=34, find k .

Answer 1

$\bigstar$ Answer

• k = ± 9.69

$\bigstar$ Given

• α and β are the zeroes of quadratic polynomial x² - kx + 15
• such that  ( α² + β² ) - 2αβ = 34 .

$\bigstar$ To Find

• Value of k

$\bigstar$ Solution

Given quadratic polynomial is x² - kx + 15

Compare given equation with ax² + bx + c , we get ,

• a = 1 , b = - k , c = 15

Also given α , β are zeroes of the quadratic equation .

Now ,

$\rm Sum\ of\ zeroes\ ,\alpha +\beta =\dfrac{-b}{a}\\\\\implies \rm \alpha +\beta =\dfrac{-(-k)}{1}\\\\\implies \rm \alpha +\beta =k...(1)\\\\\rm Product\ of\ zeroes\ ,\alpha .\beta =\dfrac{c}{a}\\\\\implies \rm \alpha .\beta =\dfrac{15}{1}\\\\\implies \rm \alpha .\beta =15...(2)$

It is also given that ,

$\alpha^2 + \beta^2 - 2 \alpha \beta = 34\\\\\implies \rm (\alpha +\beta)^2-2(\alpha.\beta)-2(\alpha.\beta)=34\\\\\implies \rm (k)^2-4(15)=34\;\;[\; From\ (1) \& (2)\;]\\\\\implies \rm k^2-60=34\\\\\implies \rm k^2=34+60\\\\\implies \rm k^2=94\\\\\implies \rm k^2=\pm \sqrt{94}\\\\\implies \bf k= \pm\ 9.69$

More Info

$\rm For\ a\ cubic\ polynomial\ ,ax^3+bx^2+cx+d,\\\\\bullet \;\; \rm Sum\ of\ zeroes\ , \alpha +\beta +\gamma =\dfrac{-b}{a}\\\\\bullet \;\; \rm Product\ of\ two\ zeroes\ ,\alpha \beta +\beta \gamma +\gamma \alpha =\dfrac{c}{a}\\\\\bullet \;\; \rm Product\ of\ zeroes\ ,\alpha \beta \gamma =\dfrac{-d}{a}$