Multiple choice

For what values of m, the equation $2x^2-2(2m+1)x+m(m+1)=0. m \in R$ has

  1. Both roots smaller than $27$
  2. Both roots greater than $27$

  3. Both roots lie in the internal (2,3)2

  4. none of these
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A Correct answer
AI explanation

We find the roots of the quadratic equation 2x^2 - 2(2m+1)x + m(m+1) = 0 by using the quadratic formula. The discriminant is [2(2m+1)]^2 - 8m(m+1) = 4m^2 + 4, so the roots evaluate to x = 0.5(2m+1 ± sqrt(m^2+1)). Because the expression sqrt(m^2+1) is strictly less than m+1 for all real m, both the positive and negative roots always evaluate to less than m+1.5, showing that both roots are consistently smaller than 27.

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