Question

If Alpha and beta are the roots of 4x^2+7x+2=0,, then the equation whose roots are Alpha^2 and beta^2is ​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{ \alpha and \beta are roots of 4x^2+7x+2=0 }$

$\underline{\textsf{To find:}}$

$\textsf{The equation whose roots are {\alpha}^2 and {\beta}^2 }$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{4x^2+7x+2=0}$

$\textsf{Then,}$

$\mathsf{\alpha+\beta=\dfrac{-7}{4}}$

$\mathsf{\alpha\,\beta=\dfrac{2}{4}=\dfrac{1}{2}}$

$\textsf{We form a quadratic equation having roots {\alpha}^2 and {\beta}^2 }$

$\textsf{Sum of the roots}$

$\mathsf{={\alpha}^2+{\beta}^2}$

$\mathsf{=(\alpha+\beta)^2-2\,\alpha\,\beta}$

$\mathsf{=(\dfrac{-7}{4})^2-2(\dfrac{1}{2})}$

$\mathsf{=\dfrac{49}{16}-1}$

$\mathsf{=\dfrac{33}{16}}$

$\textsf{Product of the roots}$

$\mathsf{={\alpha}^2\,{\beta}^2}$

$\mathsf{=(\alpha\beta)^2}$

$\mathsf{=(\dfrac{1}{2})^2}$

$\mathsf{=\dfrac{1}{4}}$

$\textsf{The required quadratic equation is}$

$\mathsf{x^2-({\alpha}^2+{\beta}^2)x+{\alpha}^2{\beta}^2=0}$

$\mathsf{x^2-\dfrac{33}{16}x+\dfrac{1}{4}=0}$

$\implies\boxed{\mathsf{16x^2-33x+4=0}}$