Question

If alpha,beta are the roots of x2 + 3x = 5 , calculate the value of alpha raise to power 4 plus beta raise to power 4.

Answer 1

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\alpha^{4}+\beta^{4}=\frac{81}{8}}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\underline \bold{Given : } \\ \implies \alpha \: and \: \beta \in( {x}^{2} + 3x - 5 = 0) \\ \\ \underline \bold{To \: Find : } \\ \implies { \alpha }^{4} + { \beta }^{4} = ?$

• According to given question :

$\bold{Using \: quadratic \: formula : } \\ \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \implies x = \frac{ - 3 \pm \sqrt{ {3}^{2} - 4 \times 1 \times ( - 5) } }{2 \times 1} \\ \\ \implies x = \frac{ - 3 \pm \sqrt{9 + 20} }{2} \\ \\ \implies x = \frac{ - 3 \pm \sqrt{29} }{2} \\ \\ \bold{\implies \alpha = \frac{ - 3 + \sqrt{29} }{2} } \\ \\ \bold{\implies \beta = \frac{ - 3 - \sqrt{29} }{2} } \\ \\ \bold{For \: finding \: value : } \\ \implies { \alpha }^{4} + { \beta }^{4} = ({ \frac{ - 3 + \sqrt{29} }{2} })^{4} + ({ \frac{ - 3 - \sqrt{29} }{2} })^{4} \\ \\ \implies { \alpha }^{4} + { \beta }^{4} = \frac{ ({ - 3})^{4} + ({ \sqrt{29} })^{4} }{ {2}^{4} } + \frac{ ({ - 3})^{4} - ({ \sqrt{29} })^{4} }{ {2}^{4} } \\ \\ \implies{ \alpha }^{4} + { \beta }^{4} = \frac{81 +29 \times 29 }{16} + \frac{81 - 29 \times 29}{16} \\ \\ \implies { \alpha }^{4} + { \beta }^{4} = \frac{81 \cancel{+ 841} + 81 \cancel{- 841}}{16} \\ \\ \implies { \alpha }^{4} + { \beta }^{4} = \frac{162}{16} \\ \\ \bold{ \implies { \alpha }^{4} + { \beta }^{4} = \frac{81}{8} }$

Answer 2

$\huge\sf{Answer:-}$

Given Question:-

If alpha,beta are the roots of x2 + 3x = 5 , calculate the value of alpha raise to power 4 plus beta raise to power 4.

Find:-

We have to find what is α^4 + β^4

$\bf{By \:Using \:Quadratic\: Formula}$

$\sf x = \frac{ - b± \sqrt{b {}^{2} - 4ac} }{2a}$

$\sf x = \frac{ - 3± \sqrt{3 {}^{2} - 4 \times 1 \times ( - 5) } }{2 \times 1}$

$\sf x = \frac{3± \sqrt{9 + 20} }{2}$

$\sf x = \frac{ - 3± \sqrt{29} }{2} = \beta = \frac{ - 2 - \sqrt{29} }{2}$

Adding Values:-

$\sf = \alpha {}^{4} + \beta {}^{4} = ( \frac{ - 3 + \sqrt{29} }{2}) {}^{4} + ( \frac{ - 3 - \sqrt{29} }{2} ) {}^{4}$

$\sf = \alpha {}^{4} + \beta {}^{4} = ( \frac{ (- 3) {}^{4} + ( \sqrt{29) {}^{4} } }{2 {}^{4} } ) +( \frac{ - 3 - \sqrt{29) {}^{4} } }{2 {}^{4} }$

$\sf = \alpha {}^{4} + \beta {}^{4} = \frac{(8 + 29 \times 29)}{16} + \frac{(18 - 29 \times 29)}{16}$

$\sf = \alpha {}^{4} + \beta {}^{4} = \frac{(81 + 841 + 81 - 841)}{16}$

$\sf = \alpha {}^{4} + b {}^{4} = \frac{162}{16}$

$\sf = \alpha {}^{4} + \beta {}^{4} = \frac{81}{8}$

$\sf\color{red}{Nayan}$$\sf\color{blue}{Shreyas ....}$