Question

if alpha and beta are the zeroes of a quadratic polynomial xsquare+5x-5 , then
a) alpha+beta = alphabeta
b) alpha-beta = alphabeta
c) alpha+beta > alphabeta
d) alpha+beta < alphabeta

Answer 1

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✦ Required Answer:

♦️ GiveN:

• $\large{\rm{\alpha}}$ and $\large{\rm{\beta}}$ are the zeroes of the polynomial $\large{\rm{{x}^{2}+5x-5}}$

♦️ To FinD:

• Relation between the $\large{\rm{\alpha}}$ + $\large{\rm{\beta}}$ and $\large{\rm{\alpha}}$ $\large{\rm{\beta}}$

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✦ How to Solve?

Before solving this question, we need to know the relationship of zeroes and coefficients of the quadratic polynomial. The relations are:

$\large{ \boxed{ \rm{ \purple{Sum \: of \: zeroes \: = - \normalsize{\frac{Coefficient \: of \: x}{Coefficient \: of \: {x}^{2} } }}}}}$

And,

$\large{ \boxed{ \rm{ \purple{Product \: of \: zeroes = \normalsize{ \frac{Constant \: term}{Coefficient \: of \: {x}^{2} }}}}}}$

I think, we should always remember this relation. I know that people often use, sum of zeroes = -b/a and product of zeroes = c/a when the quadratic polynomial ax^2+bx+c, but this might cause confusion when the coefficients are changed.

So, let's solve this question,

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✦ Solution:

Given, quadratic polynomial = $\large{\rm{{x}^{2}+5x-5}}$ and the zeroes of the quadratic polynomial are $\large{\rm{\alpha}}$ and $\large{\rm{\beta}}$.

By using relation,

$\large{ \rm{Sum \: of \: zeroes...}} \\ \large{ \rm{ \longrightarrow \: \alpha + \beta = - \frac{5}{1}}} \\ \\ \large{ \rm{ \longrightarrow \: \alpha + \beta = \boxed{ \rm{ \green{- 5}}}}} \\ \\ \large{ \rm{Product \: of \: zeroes....}} \\ \large{ \rm{ \longrightarrow \: \alpha \beta = \frac{ - 5}{1}}} \\ \\ \large{ \rm{ \longrightarrow \: \alpha \beta = \boxed{ \rm{ \green{- 5}}}}}$

So, from here we can conclude that,

$\large{ \rm{ \longrightarrow \: \alpha + \beta = \alpha \beta }} \\ \\ \rm{\underline{ \dag \: { \pink{As \: both \: are \: equal \: to - 5}}}}$

✏Thus, Option A is correct.

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Answer 2

$\purple{\underline{\underline{\pink{\boxed{\boxed{\red{\star{\sf Question:}}}}}}}}$

$\rm{If \ \alpha \ and \ \beta \ are \ the \ zeroes \ of \ a}$

$\rm{quadratic \ polynomial \ x^2+5x-5, \ then}$

$a) \: \: \: \alpha + \beta = \alpha \beta$

$b) \: \: \alpha - \beta \alpha \beta$

$c) \: \: \alpha + \beta > \alpha \beta$

$d) \: \: \alpha + \beta < \alpha \beta$

_________________________________________

$\purple{\underline{\underline{\orange{\boxed{\boxed{\green{\star{\sf Answer:}}}}}}}}$

$\rm\orange{The \ correct \ option \ is :}$

$\rm\orange{a) \: \: \: \alpha + \beta = \alpha \beta }$

_________________________________________

$\sf\large\underline\pink{Given:}$

$\rm{The \ given \ quadratic \ polynomial \ is }$

${x}^{2} + 5x - 5$

_________________________________________

$\sf\large\underline\blue{To \ Find}$

$\rm{correct \ option}$

_________________________________________

$\green{\underline{\underline{\red{\boxed{\boxed{\purple{\star{\sf Solution:}}}}}}}}$

$\rm{Given, \ quadratic \ polynomial}$

$\implies\rm{{x}^{2}+5x-5}$

$\rm{the \ zeroes \ of \ the \ quadratic \ polynomial }$

$\implies\rm{\alpha \ and \ \beta}$.

$\rm{Now \ using}$

$\rm{Sum \: of \: zeroes...}$

$\rm\implies{ \alpha + \beta = - \frac{5}{1}}$

$\rm\therefore{\boxed{\orange{ \alpha + \beta \ = \ -5}}}$

$\rm{Product \: of \: zeroes....}$ $\rm\therefore{\alpha \beta = \frac{ - 5}{1}}$

$\rm\therefore{\boxed{\orange{ \alpha \beta \ = \ -5}}}$

$\rm{Hence \ we \ get ,}$

$\bf\implies{\blue{\boxed{\red{\alpha + \beta = \alpha \beta}}}}$

$\large\tt\purple{Option \ a) \ is \ correct}$