4a-2-19a-5-x-2-a-2-x-a-3-0-x-1-x-2-are-roots-when-x-1-lt-0-x-2-gt-0-x-1-x-2-gt-0-interval-of-max-a-solution-

Question Number 190731 by 073 last updated on 10/Apr/23

({4 a}^{2} - 19 a - 5) x^{2} + a^{2} x + a + 3 = 0

x_{1}, x_{2} are roots

when, x_{1} < 0, x_{2} > 0, ∣ x_{1} ∣ - x_{2} > 0

interval of \max (a) = ?

solution ? ?

Answered by 073 last updated on 10/Apr/23

Answered by manxsol last updated on 10/Apr/23

x_{1} < 0, x_{2} > 0 ∣ x_{1} ∣ - x_{2} > 0

\Rightarrow - x_{1} - x_{2} > 0

x_{1} + x_{2} < 0

- \frac{b}{a} < 0

- \frac{a^{2}}{(4 a^{2} - 19 a - 5)} < 0

\frac{a^{2}}{(4 a + 1) (a - 5)} < 0

a \neq 0

p t o s c r i t i c o s

+ (- \frac{1}{4}) - (5) +

a ϵ < - \frac{1}{4}, 5 > - {0} A

x_{1} < 0 x_{2} > 0 \Rightarrow x_{1} x_{2} < 0

\frac{(a + 3)}{(4 a + 1) (a - 5)} < 0

- (- 3) + (- \frac{1}{4}) - (5) +

< - \infty, - 3 > \cup < - \frac{1}{4}, 5 > B

A \cap B =< - \frac{1}{4}, 5 > - {0}

Commented by 073 last updated on 10/Apr/23

nice solution

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