Question

Given that a=alpha and b=beta are roots of 2x^2+x+14=0 find the equation whose roots are a^4 and b^4

Answer 1

$\textbf{Given:}$

$\text{a and b are roots of 2x^2+x+14=0 }$

$\textbf{To find:}$

$\text{The quadraatic equation whose roots are a^4 and b^4 }$

$\textbf{Solution:}$

$\text{Consider,}\;2x^2+x+14=0$

$\text{Then,}$

$a+b=\dfrac{-1}{2}$

$a\,b=\dfrac{14}{2}=7$

$a^2+b^2=(a+b)^2-2\,ab$

$a^2+b^2=(\frac{-1}{2})^2-2(7)$

$a^2+b^2=\frac{1}{4}-14$

$a^2+b^2=\dfrac{1-56}{4}=\frac{-55}{4}$

$\bf\,a^4+b^4$

$=(a^2+b^2)^2-2\,a^2b^2$

$=(\frac{-55}{4})^2-2(7)^2$

$=\frac{3025}{16}-98$

$=\frac{3025-1568}{16}$

$=\frac{1457}{16}$

$\bf\,a^4\,b^4$

$=(a\,b)^4$

$=(7)^4$

$=2401$

$\text{The required quadratic equation}$

$x^2-(a^4+b^4)x+a^4b^4=0$

$x^2-(\frac{1457}{16})x+2401=0$

$\implies\boxed{\bf\,16x^2-1457\,x+38416=0}$

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