Question

Answer these questions
plz answer 10,11,12

Answer 1

Heya !!

Here's your answer..⬇⬇
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Q.10

p(x) = x² - ( k - 6 )x + ( 2k + 1 )
a = 1
b = - ( -k - 6 )
c = ( 2k + 1 )

$sum \: \: of \: \: root \: = \frac{ - b}{a} \\ \\ \alpha + \beta = \frac{ - ( - k + 6)}{1} \\ \\ \alpha + \beta = k - 6 \\ \\ product \: \: of \: \: zeros \: = \frac{c}{a} \\ \\ \alpha \beta = 2k + 1 \\ \\ given \: \: that \: \: \alpha + \beta = \alpha \beta \\ \\ k - 6 = 2k + 1 \\ \\ k - 2k = 7 \\ \\ k = - 7 \\ = = = = = = = = = = = = = = = = \\ p(y) = {y}^{2} - 8y + a \\ \\ a = 1 \: \: \: b = ( - 8) \: \: \: c = a \\ \\ sum \: \: of \: \: roots \: = \frac{ - b}{a} \\ \\ \alpha + \beta = \frac{ - ( - 8)}{1} \\ \\ \alpha + \beta = 8 \\ \\ product \: \: of \: \: roots \: = \frac{c}{a} \\ \alpha \beta = a \\ \\ take \: \: square \: \: on \: ( \alpha + \beta ) \\ \\ {( \alpha + \beta )}^{2} = {( \alpha )}^{2} + {( \beta )}^{2} + 2 \alpha \beta \\ \\ {(8)}^{2} = 40 + 2( a) \\ \\ 64 = 40 + 2a \\ \\ 2a = 24 \\ \\ a = 12 \\ = = = = = = = = = = f(x) = 3 {x}^{2} - 6x + 4 \\ \\ a = 3 \: \: \: b = ( - 6) \: \: \: c = 4 \\ \\ sum \: \: of \: \: roots \: = \frac{ - b}{a} \\ \\ \alpha + \beta = \frac{ - ( - 6)}{3} \\ \\ \alpha + \beta = 2 \\ \\ product \: \: of \: \: zeros \: = \frac{c}{a} \\ \\ \alpha \beta = \frac{4}{3} \\ \\now.. \\ = \frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } + 2( \frac{1}{ \alpha } + \frac{1}{ \beta } ) + 3 \alpha \beta \\ \\ = \frac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta } + 2( \frac{ \beta + \alpha }{ \alpha \beta } ) + 3 \alpha \beta \\ \\ = \frac{ {( \alpha + \beta )}^{2} }{ \alpha \beta } + 2( \frac{ \beta + \alpha }{ \alpha \beta } ) + 3 \alpha \beta \\ \\ = \frac{ {(2)}^{2} }{ \frac{4}{3} } + 2( \frac{2}{ \frac{4}{3} } ) + 3( \frac{4}{3} ) \\ \\ = \frac{12}{4} + \frac{12}{4} + \frac{12}{3} \\ \\ = 3 + 3 + 4 \\ \\ = 10$
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Hope it helps..
Thanks :)

Answer 2

Hey☺☺
Your answer is here

1)Alpha + beta=-b/a

Alphabeta=c/a

If alpha and beta are quadratic polynomial alpha+beta=alphabeta

given:-
p(x)=x^2-(k-6)x+(2k+1)

Let alpha + beta be (k-6)
Let alphabeta be (2k+1)

given:-
alpha +beta=alphabeta
k-6=2k+1
k-2k=1+6
-k=7
k=-7

2)Given:-
p(x)=y^2-8y+a
alpha+beta=-b/a
a=1
b=-8
c=a

alpha + beta=-b/a
=8
alphabeta=a
(alpha+beta)^2=alpha^2+beta^2+2alphabeta
8^2=40+2a
64=40+2a
64-40=2a
24=2a
24/2=a
a=12

hope it helps☺☺☺