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

(4a^2 −19a−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??

$$\left(\mathrm{4a}^{\mathrm{2}} −\mathrm{19a}−\mathrm{5}\right)\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} \mathrm{x}+\mathrm{a}+\mathrm{3}=\mathrm{0} \\ $$ $$\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} \mathrm{are}\:\mathrm{roots} \\ $$ $$\mathrm{when}\:,\:\mathrm{x}_{\mathrm{1}} <\mathrm{0}\:\:\:,\mathrm{x}_{\mathrm{2}} >\mathrm{0}\:\:,\:\mid\mathrm{x}_{\mathrm{1}} \mid−\mathrm{x}_{\mathrm{2}} >\mathrm{0} \\ $$ $$\mathrm{interval}\:\mathrm{of}\:\:\:\mathrm{max}\left(\mathrm{a}\right)=? \\ $$ $$\mathrm{solution}?? \\ $$

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 ⇒−x_1 −x_2 >0 x_1 +x_2 <0 −(b/a)<0 −(a^2 /((4a^2 −19a−5)))<0 (a^2 /((4a+1)(a−5)))<0 a≠0 ptos criticos + (−(1/4)) − (5) + a ε <−(1/4),5>−{0} A x_1 <0 x_2 >0 ⇒x_1 x_2 <0 (((a+3))/((4a+1)(a−5)))<0 − (−3)+ (−(1/4)) − (5) + <−∞,−3>∪<−(1/4),5> B A∩B=<−(1/4),5>−{0}

$${x}_{\mathrm{1}} <\mathrm{0},\:{x}_{\mathrm{2}} >\mathrm{0}\:\:\:\:\:\:\mid{x}_{\mathrm{1}} \mid−{x}_{\mathrm{2}} >\mathrm{0} \\ $$ $$\Rightarrow−{x}_{\mathrm{1}} −{x}_{\mathrm{2}} >\mathrm{0} \\ $$ $${x}_{\mathrm{1}} +{x}_{\mathrm{2}} <\mathrm{0} \\ $$ $$−\frac{{b}}{{a}}<\mathrm{0} \\ $$ $$−\frac{{a}^{\mathrm{2}} }{\left(\mathrm{4}{a}^{\mathrm{2}} −\mathrm{19}{a}−\mathrm{5}\right)}<\mathrm{0} \\ $$ $$\frac{{a}^{\mathrm{2}} }{\left(\mathrm{4}{a}+\mathrm{1}\right)\left({a}−\mathrm{5}\right)}<\mathrm{0} \\ $$ $${a}\neq\mathrm{0} \\ $$ $${ptos}\:{criticos} \\ $$ $$\:\:\:\:\:\:\:+\:\:\:\:\:\:\left(−\frac{\mathrm{1}}{\mathrm{4}}\right)\:\:\:\:\:\:−\:\:\:\:\:\left(\mathrm{5}\right)\:\:\:\:\:\:+ \\ $$ $${a}\:\epsilon\:\:\:<−\frac{\mathrm{1}}{\mathrm{4}},\mathrm{5}>−\left\{\mathrm{0}\right\}\:\:\:\:\:\:{A} \\ $$ $${x}_{\mathrm{1}} <\mathrm{0}\:{x}_{\mathrm{2}} >\mathrm{0}\:\:\:\:\Rightarrow{x}_{\mathrm{1}} {x}_{\mathrm{2}} <\mathrm{0} \\ $$ $$\frac{\left({a}+\mathrm{3}\right)}{\left(\mathrm{4}{a}+\mathrm{1}\right)\left({a}−\mathrm{5}\right)}<\mathrm{0} \\ $$ $$−\:\left(−\mathrm{3}\right)+\:\:\:\:\:\left(−\frac{\mathrm{1}}{\mathrm{4}}\right)\:\:\:\:\:−\:\:\:\:\left(\mathrm{5}\right)\:\:\:+ \\ $$ $$<−\infty,−\mathrm{3}>\cup<−\frac{\mathrm{1}}{\mathrm{4}},\mathrm{5}>\:\:{B} \\ $$ $${A}\cap{B}=<−\frac{\mathrm{1}}{\mathrm{4}},\mathrm{5}>−\left\{\mathrm{0}\right\}\:\: \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$

Commented by073 last updated on 10/Apr/23

nice solution

$$\mathrm{nice}\:\mathrm{solution} \\ $$

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