Question

Rezolvați exercițiul în spațiul indicat:
Diferența soluțiilor ecuației x2+5x+m=0 este 9. Aflați soluțiile și m.

Answer 1

 

$\displaystyle\bf\\Se~da:\\Ecuatia: x^2+5x+m=0\\x_1-x2=9\\\\Se~cere:\\x_1=?\\x_2?\\m=?\\\\Rezolvare:\\\\x^2+5x+m=0\\\\x_{12}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-5\pm\sqrt{25-4m}}{2}\\\\x_1=\frac{-5+\sqrt{25-4m}}{2}\\\\x_2=\frac{-5-\sqrt{25-4m}}{2}\\\\x_1-x2=9\\\\\frac{-5+\sqrt{25-4m}}{2}-\frac{-5-\sqrt{25-4m}}{2}=9~~\Big|\times 2\\\\\Big(-5+\sqrt{25-4m}\Big)-\Big(-5-\sqrt{25-4m}\Big)=18$

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$\displaystyle\bf\\Desfacem~parantezele:\\\\-5+\sqrt{25-4m}+5+\sqrt{25-4m}=18\\\\\underbrace{-5+5}_{=0}+\sqrt{25-4m}+\sqrt{25-4m}=18\\\\2\sqrt{25-4m}=18~~\Big|:2\\\\\sqrt{25-4m}=9~~\Big|~ridicam~la~a~2-a\\\\25-4m=81\\\\4m=25-81\\\\4m=25-81\\\\4m=-56\\\\m=-\frac{56}{4}\\\\\boxed{\bf m=-14}$

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$\displaystyle\bf\\Calculam~solutiile:\\\\x_{12}=\frac{-5\pm\sqrt{25-4m}}{2}\\\\m=-14\\\\x_{12}=\frac{-5\pm\sqrt{25-4\times(-14)}}{2}\\\\x_{12}=\frac{-5\pm\sqrt{25+56}}{2}\\\\x_{12}=\frac{-5\pm\sqrt{81}}{2}\\\\x_{12}=\frac{-5\pm9}{2}\\\\x_1=\frac{-5+9}{2}=\frac{4}{2}=2\\\\x_1=\frac{-5-9}{2}=\frac{-14}{2}=-7\\\\\boxed{\bf x_1=2}\\\\\boxed{\bf x_2=-7}\\\\Verificare:\\\\x1-x2=2-(-7)=2+7=9~~(corect)$

 

 

 

Answer 2

Răspuns:

Poate ceva mai simplu, folosind relatiile lui Viete

Ptr ax^2+bx+c=0

X1+X2= - b/a

X1*X2=c/a

Explicație pas cu pas:

Prin comparatie cu forma generala obtinem

a=1

b=5

c=m

X1 +X2= - 5/1= - 5

X1-X2=9

2X1=4

X1=2

X2= - 5-X1= - 7

ULTIMA RELATIE A LUI VIETE

X1*X2=c=m

2*(-7)=m

m= - 14