Question

Să se rezolve ecuațiile:g).2mx^2+(1-2m^2)x-m=0

Answer 1

   
[tex]\displaystyle\\ \text{coeficientii ecuatiei sunt:}\\\\ a=2m\\ b=(1-2m^2)\\ c=-m\\\\ x_{12}= \frac{-b\pm \sqrt{b^2-4ac} }{2a} =\\\\\\ =\frac{-(1-2m^2)\pm \sqrt{(1-2m^2)^2-4\cdot 2m \cdot (-m)} }{2\cdot 2m} = \\\\\\ =\frac{2m^2-1\pm \sqrt{1-4m^2+4m^4+8m^2} }{4m} = \\\\\\ =\frac{2m^2-1\pm \sqrt{1+4m^2+4m^4} }{4m} =\\\\\\ =\frac{2m^2-1\pm \sqrt{(1+2m^2)^2} }{4m} =\\\\\\ =\frac{2m^2-1\pm (1+2m^2) }{4m} [/tex]


[tex]\displaystyle\\ x_1=\frac{2m^2-1+ (1+2m^2) }{4m} =\frac{2m^2-1+ 1+2m^2 }{4m} =\frac{4m^2 }{4m} =\boxed{\bf m}\\\\ x_2=\frac{2m^2-1- (1+2m^2) }{4m} =\frac{2m^2-1-1-2m^2 }{4m} =\frac{-2}{4m} =\boxed{\bf -\frac{1}{2m}} [/tex]