Solved: "2a2−bc−9a+10=0 then the range of ‘a' is?Let a,..."

diartrosiwk8o

Answered question

2022-03-31

2 a^{2} - b c - 9 a + 10 = 0

then the range of ‘a' is?
Let a, b and c be real numbers such that

2 a^{2} - b c - 9 a + 10 = 0

and

4 b^{2} + c^{2} + b c - 7 a - 8 = 0

, then the range of ‘a' is?

glikozyd3s68

Beginner2022-04-01Added 16 answers

From the first equality

b c = 2 a^{2} - 9 a + 10

\Rightarrow (4 b) c = 8 a^{2} - 36 a + 40 (1)

Substituting in the 2nd expression,

4 b^{2} + c^{2} - 9 a + 2 a^{2} + 10 - 7 a - 8 = 0

\Rightarrow 4 b^{2} + c^{2} = - 2 a^{2} + 16 a + 8 (2)

Adding (1) and (2), we get

{(2 b + c)}^{2} = 6 a^{2} - 20 a + 48 = 2 (3 a^{2} - 10 a + 24)

Note that

3 a^{2} - 10 a + 24 > 0, \forall a

Similarly on subtracting (1) from (2), we get

{(2 b - c)}^{2} = - 10 a^{2} + 52 a - 32 = - 2 (5 a^{2} - 26 a + 16)

Find the range of a where

5 a^{2} - 26 a + 16 \leq 0

This range will give the answer

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