Multiple choice

If the equation $(m-n)x^2+(n-l)x+l-m=0$ has equal roots, then l, m and n satisfy.

  1. $2l=m+n$
  2. $2m=n+l$
  3. $m=n+l$
  4. $l=m+n$
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B Correct answer
Explanation

For a quadratic equation Ax^2 + Bx + C = 0 to have equal roots, the discriminant B^2 - 4AC must be 0. Here A=(m-n), B=(n-l), C=(l-m). Substituting these, (n-l)^2 - 4(m-n)(l-m) = 0 leads to (n-l)^2 + 4(m-n)(m-l) = 0, which simplifies to (n+l-2m)^2 = 0, so 2m = n+l.

AI explanation

For the quadratic equation to have equal roots, its discriminant must be zero, so (n - l)^2 - 4(m - n)(l - m) = 0. Expanding this gives n^2 + l^2 - 2nl - 4(ml - m^2 - nl + nm) = 0, which simplifies to n^2 + l^2 + 2nl - 4ml + 4m^2 - 4nm = 0. This expression can be factored as (n + l - 2m)^2 = 0. Taking the square root of both sides yields n + l = 2m.

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