Question

If alpha and beta are the zeros of the quadratic polynomial p(x)=x^2+12x+35,form a quadratic polynomial whose zeros are 2alpha,2beta

Answer 1

Given that, α and β are the zeros of the quadratic polynomial p(x) = x² + 12x + 35.

We've to form a Quadratic Polynomial whose Zeroes are 2α and 2β.

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$\bigstar\:{\underline{\pmb{\sf{\purple{Given\:Polynomial\::\:}x^2 + 12x + 35}}}}$

Here,

• a = 1 ; b = 12 and c = 35

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As we know that,

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$\qquad\bf{\dag} \: {\underline{\pmb{\sf{Sum \: of \: Zeroes = \red{- b/a}}}}}\\\\\\ \qquad\dashrightarrow\sf \alpha + \beta = \dfrac{-12}{1}\\\\\\ \qquad\dashrightarrow{\underline{\boxed{\frak{ \pmb{\alpha + \beta = \purple{- 12}}}}}}$

$\qquad\bf{\dag} \: {\underline{\pmb{\sf{Product \: of \: Zeroes = \red{c/a}}}}}\\\\\\ \qquad\dashrightarrow\sf \alpha \beta = \dfrac{35}{1}\\\\\\ \qquad\dashrightarrow{\underline{\boxed{\frak{ \pmb{\alpha \beta = \purple{35}}}}}}$

☆ Now, Let's find the sum and product of Zeroes of new Quadratic Polynomial :

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$\qquad\bf{\dag} \: \: {\underline{\pmb{\sf{Sum \: of \: Zeroes = \red{2 \alpha + 2 \beta }}}}}\\\\\\\qquad \dashrightarrow\sf 2(\alpha + \beta)\\\\\\ \qquad\dashrightarrow\sf 2( - 12)\\\\\\ \qquad\dashrightarrow{\boxed{\boxed{\frak{ \pmb{\blue{\quad - 24\quad}}}}}}$

$\qquad\bf{\dag} \: \: {\underline{\pmb{\sf{Product \: of \: Zeroes = \red{2 \alpha \times 2 \beta }}}}}\\\\\\ \qquad\dashrightarrow\sf 2(\alpha \times \beta)\\\\\\ \qquad\dashrightarrow\sf 2( 35)\\\\\\ \qquad\dashrightarrow{\boxed{\boxed{\frak{ \pmb{ \blue{\quad70\quad}}}}}}$

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☆ Now, By using Sum and Product of Zeroes Let's find out New required Quadratic Polynomial by using Formula :

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$\maltese \: { \underline{\boxed{\sf{p(x) = x^2 - \bigg(\blue{Sum\:of\:Zeroes} \bigg)x + \blue{Product\:of\:zeroes}}}}}\\\\\\ \dashrightarrow\sf p(x) = x^2 - (- 24)x + 70\\\\\\ \dashrightarrow{\underline{\boxed{\frak{ \pmb{\pink{p(x) = x^2 + 24x + 70}}}}}} \:\bigstar$

$\therefore\:{\underline{\sf{Hence,\:the\:required\:Quadratic\:Polynomial\:is\: \pmb{x^2 + 24x + 70}.}}}$