Question

if alpha beta are the zeroes of x^ -2x -1 form a quadratic polynomial whose zeroes are 2alpha -1 and 2beta -1​

Answer 1

Answer:

,

Step-by-step explanation:

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Answer 2

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

If α and β are the zeroes of x² - 2x - 1 form a quadratic polynomial whose zeroes are 2α - 1 and 2β - 1.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

A quadratic polynomial.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We have p(x) = x² - x - 1

Zero of the polynomial p(x) = 0

So;

As we know that given polynomial compared with ax² + bx + c

• a = 1
• b = -2
• c = -1

$\underline{\pink{\mathcal{SUM\:OF\:ZEREOS\::}}}$

$\longrightarrow\sf{\alpha +\beta =\dfrac{-b}{a} }\\\\\\\longrightarrow\sf{\alpha +\beta =\dfrac{-(-2)}{1} }\\\\\\\longrightarrow\bf{\alpha +\beta =2}}$

$\underline{\pink{\mathcal{PRODUCT\:OF\:ZEREOS\::}}}$

$\longrightarrow\sf{\alpha \times \beta =\dfrac{c}{a} }\\\\\\\longrightarrow\sf{\alpha \times \beta =\dfrac{-1}{1} }\\\\\\\longrightarrow\bf{\alpha \times \beta =-1}}$

A/q

$\dag\underline{\underline{\bf{Sum\:of\:the\:zeroes\::}}}}}$

$\longrightarrow\sf{2\alpha \cancel{-1}+2\beta \cancel{-1}}\\\\\\\longrightarrow\sf{2\alpha +2\beta }\\\\\\\longrightarrow\sf{2(\alpha +\beta )}\\\\\\\longrightarrow\sf{2(2)}\\\\\\\longrightarrow\sf{\orange{4}}$

$\dag\underline{\underline{\bf{Product\:of\:the\:zeroes\::}}}}}$

$\longrightarrow\sf{(2\alpha -1)(2\beta -1)}\\\\\\\longrightarrow\sf{4\alpha \beta -2\alpha -2\beta +1}\\\\\\\longrightarrow\sf{4\alpha \beta -2(\alpha +\beta )+1}\\\\\\\longrightarrow\sf{4(-1)-2(2)+1}\\\\\\\longrightarrow\sf{\cancel{-4-4} +1}\\\\\\\longrightarrow\sf{\orange{1}}$

Thus;

$\dag\underline{\underline{\bf{The\:required\:quadratic\:polynomial\::}}}}}$

$\mapsto\sf{x^{2} -(sum\:of\:zeroes)x+(product\:of\:zeroes)}\\\\\mapsto\sf{x^{2} -(4)x+1}\\\\\mapsto\sf{\orange{x^{2} -4x+1=0}}$