Question

if p is real, what is the nature of the roots of x^2+2(p+1)x+2p=0?​

Answer 1

Step-by-step explanation:

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$\large\underline{\underline\bold\red{Question :- }}$

If p is real, then what is the nature of the roots in the given equation :-

$\large{\bold{→ {x}^{2} + 2 \: (p + 1)x + 2p }}$

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$\large\underline{\underline\bold\red{Answer→}}$

$\large{\bold{➡ \: {x}^{2} + 2 \: (p + 1)x + 2p }}$

$\large{\bold{➡ \: (x + p + 1) {}^{2} + 2p - (p + 1) {}^{2} }}$

$\large{\bold{➡ (x + p + 1) {}^{2} - {p}^{2} - 1 }}$

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$\bigstar\large\underline{\bold\pink{Consequently, \: the \: roots \: are \: both \: real, \: and \: distinct.}}$

$\large{\bold{➡ \: x = - p - 1 ± \sqrt{ {p}^{2} + 1 } }}$

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$\bigstar \: \large\underline{\bold\blue{Extra \: Information :- }}$

There are always two real roots and at least one will be negative ( the one with the minus sign ). Both will be negative only if p is positive. If p=0 the roots are 0 and −2.

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$\huge\colorbox{pink}{Hope lt'z Help You ❥ }$