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If the roots m and n(m+n is not equal to zero) of the quadratic equation ax2+bx+c=0 are real and of opposite sign.Then show that the roots of the equation m(x-n)2 + n(x-m)2=0 are also real and of opposite sign.

If the roots m and n(m+n is not equal to zero) of the quadratic equation ax2+bx+c=0 are real and of opposite sign.Then show that the roots of the equation m(x-n)2 + n(x-m)2=0 are also real and of opposite sign.

Grade:12th Pass

2 Answers

sanatan sharma
26 Points
9 years ago

ax2+bx+c=0

m+n=-b/a

mn=c/a

solving m(x-n)2 + n(x-m)2=0

  m(x2+n2-2xn) + n(x2+m2-2xm)=0

  mx2+mn2-2xmn + nx2+nm2-2xmn=0

  x2(m+n) -(2mn)x +mn(m+n) = 0

 -bx2/a - 2cx/a - bc/a2 =0

 -( abx2 + 2acx+bc )=0

      abx2 + 2acx+bc=0

  let its root be α & β

  α + β = -2c/b

       αβ = c/a

   α + β = 2/(m+n)

      αβ  = mn

  hence proved

 

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sanatan sharma
26 Points
9 years ago

ax2+bx+c=0

m+n=-b/a

mn=c/a

solving m(x-n)2 + n(x-m)2=0

  m(x2+n2-2xn) + n(x2+m2-2xm)=0

  mx2+mn2-2xmn + nx2+nm2-2xmn=0

  x2(m+n) -(4mn)x +mn(m+n) = 0

 -bx2/a - 4cx/a - bc/a=0

 -( abx+ 4acx+bc )=0

      abx2 + 4acx+bc=0

  let its root be α & β

  α + β = -4c/b

       αβ = c/a

   α + β = 4/(m+n)

      αβ  = mn

  hence proved

 

please approve the answer by clicking below

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