Question

$if \: \alpha \: and \: \beta \: are \: the \: zeros \: of \: polynomial \: then \: find \: polymial \: whose \: zeros \: are \: 2 \alpha + 1 \: and \: 2 \beta + 1$
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Answer 1

ANSWER :–

▪︎ ${ \bold{ {x }^{2} - 4x - 5 = 0 }}$

EXPLANATION :–

GIVEN :–

▪︎ A quadratic equation x² - x - 2 = 0 have two roots ${ \bold{ \alpha \: \: and \: \: \beta }}$

TO FIND :–

▪︎ A quadratic equation whose roots are ${ \bold{ (2 \alpha + 1) \: \: and \: \: (2 \beta + 1) }}$

SOLUTION :–

▪︎ Let's compare quadratic equation with standard equation ax² + bx + c = 0 , So that –

• a = 1 , b = -1 and c = -2

▪︎ We know that –

$\\ { \bold{ . \: \: \: Sum \: \: of \: \: \: roots \: \: = \alpha + \beta = - \dfrac{b}{a} }}$

$\\ \implies { \bold { \alpha + \beta = - \dfrac{( - 1)}{1} }}$

$\\ \implies { \bold { \alpha + \beta = 1 \: \: \: \: \: \: - - - - - eq.(1)}}$

$\\ { \bold{ . \: \: Product \: \: \: of \: \: \: roots \: \: = \alpha . \beta = \dfrac{c}{a} }}$

$\\ \implies { \bold{ \: \: \alpha . \beta = \dfrac{ - 2}{1} }}$

$\\ \implies { \bold{ \: \: \alpha . \beta = - 2 \: \: \: - - - - eq.(2) }}$

☞ Now let's find –

$\\ \implies { \bold{ \: \: \: Sum \: \: of \: \: \: roots \: \: for \: \: required \: \: equation = (2 \alpha + 1) + (2 \beta + 1) }}$

$\\ { \bold{ . \: \: \: Sum \: \: of \: \: \: roots \: \: for \: \: required \: \: equation = 2( \alpha + \beta )+2 }}$

$\\ { \bold{ . \: \: \: Sum \: \: of \: \: \: roots \: \: for \: \: required \: \: equation = 2(1)+2 }} \: \: \: \: \: [ \: using \: \: eq.(1) \:]$

$\\ { \bold{ . \: \: \: Sum \: \: of \: \: \: roots \: \: for \: \: required \: \: equation = 4 \: \: \: - - - - eq.(3)}}$

$\\ \implies { \bold{ \: \: Product \: \: \: of \: \: \: roots \: \: of \: \: required \: \: equation = (2 \alpha + 1) .( 2 \beta + 1) }}$

$\\ { \bold{ . \: \: Product \: \: \: of \: \: \: roots \: \: of \: \: required \: \: equation = 4( \alpha . \beta ) + 2( \alpha + \beta ) + 1 }}$

$\\ { \bold{ . \: \: Product \: \: \: of \: \: \: roots \: \: of \: \: required \: \: equation = 4( - 2 ) + 2( 1 ) + 1 }}$

$\\ { \bold{ . \: \: Product \: \: \: of \: \: \: roots \: \: of \: \: required \: \: equation = - 8 + 2 + 1 }}$

$\\ { \bold{ . \: \: Product \: \: \: of \: \: \: roots \: \: of \: \: required \: \: equation = - 5 \: \: \: \: - - - - eq.(4)}}$

☞ We know that a quadratic equation –

$\\ \implies { \boxed{ \bold{ {x }^{2} - (sum \: \: of \: \: roots)x + (product \: \: of \: \: roots) }}}$

▪︎ Now put the values from eq.(3) and (4) –

$\\ \implies { \boxed { \bold{ {x }^{2} - 4x - 5 = 0 }}}$