hello solve it please please
Answers
Answer:
fxjhfoeaychkyfidtisxkgdutdmvegjdsg8ye
Answer:
\red {\huge {\underline{ \frak{Your \: answEr : }}}}
YouranswEr:
\blue{\huge \implies \boxed{\beta \in(1 \: ,∞)}}⟹
β∈(1,∞)
\red {\huge {\underline{ \frak{Explanation : }}}}
Explanation:
\green{\large{ \underline{\mathcal{\star \:Given : }}}}
⋆Given:
\begin{lgathered}\tt{\alpha \: \beta \: are \: real \: number} \\ z \: \tt{is \: a \: complex \: number}\end{lgathered}
αβarerealnumber
zisacomplexnumber
\begin{lgathered}{z}^{2} + \alpha z + \beta = 0 \: \tt{has \: two \: distinct} \\ \tt{roots \: on} \: \: \mathcal{R_ez = 1}\end{lgathered}
z
2
+αz+β=0hastwodistinct
rootsonR
e
z=1
\green{\large{ \underline{\mathcal{\star \: To \: Find : }}}}
⋆ToFind:
\tt{value \: of \: \beta }valueofβ
\green{\large{ \underline{\mathcal{\star \: Solution : }}}}
⋆Solution:
\blacksquare \: \underline\frak{Quick \: Facts}■
QuickFacts
• If the roots are ax² + bx + c = 0 are Distinct and Complex, then Discriminent D < 0
• D = b² - 4ac
\blacksquare \: \underline\frak{Now \: head \: to \: the \: Question}■
NowheadtotheQuestion
★ Roots of the Equation ;
\huge z = \frac{ - \alpha ± \sqrt{ { \alpha }^{2} - 4 \beta } }{2}z=
2
−α±
α
2
−4β
★ Since z is Complex ;
\huge D = { \alpha }^{2} - 4 \beta < 0D=α
2
−4β<0
★ Also Given that ;
\huge R_ez = 1 \: , \: \frac{ - \alpha }{2} = 1R
e
z=1,
2
−α
=1
\large\implies \alpha = - 2⟹α=−2
\begin{lgathered}\large \therefore 4 - 4 \beta < 0 \\ \: \: \: \: \: \beta > 1\end{lgathered}
∴4−4β<0
β>1
\purple{\large \implies \boxed{\beta \in(1 \: ,∞)}}⟹
β∈(1,∞)
\huge{\red{\ddot{\smile}}}
⌣
¨