Question

please give a clear explanation

30 POINTS :)

Answer 1

x² - x -3m =0 let 2P and Q are the roots of this equation .

2P + Q = 1 -----(1)
2PQ = -3m-------(2)

x² -x - m =0 let P and R are the roots of this equation .

P + R = 1 ------(3)
PR = -m --------(4)

solve eqn (1) and (2)
so, 2P + Q = P + R
P + Q = R

put this in eqn 4

P ( P + Q ) = -m
P² + PQ = -m
p² -3m/2 = - m
P² = m/2

so, P²Q² = 9m²/4 eqn (2)

Q² = 9m²/4m/2 = 9m/2

now,
4P² + Q² = 4m/2 + 9m/2 = 13m/2
(2P+Q)² -4PQ = 13m/2
1² +6m = 13m/2 { from eqn (1) and (2)
m = 2


===================================
a² +2b = 7
b² +4c = -7
c² + 6a = -14

add of all equations

a² + b² + c² +2b +4c + 6a = -14

a² + 6a +9 + b² + 2b +1 + c² + 4c +4 = -14 +14

(a² +2×3a +3²) + ( b² +2b +1) + (c² +2×2c +2) =0

(a + 3)² + ( b +1)² + ( c + 2)² =0
this is possible only when ,
a+ 3 =0
a = -3
b +1 =0
b = -1
c +2 =0
c = -2
hence, a = -3, b = -1, and c = -2
.so, (a² +b² + c²)/2 = ( 9 + 1 +4 )/2 = 7

Answer 2

Hey mate here is your answer

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x² - x -3m =0 let 2P and Q are the roots of this equation .

2P + Q = 1 -----(1)

2PQ = -3m-------(2)

x² -x - m =0 let P and R are the roots of this equation .

P + R = 1 ------(3)

PR = -m --------(4)

solve eqn (1) and (2)

so, 2P + Q = P + R

P + Q = R

put this in eqn 4

P ( P + Q ) = -m

P² + PQ = -m

p² -3m/2 = - m

P² = m/2

so, P²Q² = 9m²/4 eqn (2)

Q² = 9m²/4m/2 = 9m/2

now,

4P² + Q² = 4m/2 + 9m/2 = 13m/2

(2P+Q)² -4PQ = 13m/2

1² +6m = 13m/2 { from eqn (1) and (2)

m = 2

===================================

a² +2b = 7

b² +4c = -7

c² + 6a = -14

add of all equations

a² + b² + c² +2b +4c + 6a = -14

a² + 6a +9 + b² + 2b +1 + c² + 4c +4 = -14 +14

(a² +2×3a +3²) + ( b² +2b +1) + (c² +2×2c +2) =0

(a + 3)² + ( b +1)² + ( c + 2)² =0

this is possible only when ,

a+ 3 =0

a = -3

b +1 =0

b = -1

c +2 =0

c = -2

hence, a = -3, b = -1, and c = -2

.so, (a² +b² + c²)/2 = ( 9 + 1 +4 )/2 = 7

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