Multiple choice

$x^2-(m-3)x+m=0::(m \in R)$ be a quadratic equation. Find the value of $m$ for which b oth the roots are equal:

  1. $\{1,9\}$
  2. $0$
  3. $3$
  4. $\{4,11\}$
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A Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, (m-3)^2 - 4(1)(m) = 0. m^2 - 6m + 9 - 4m = 0. m^2 - 10m + 9 = 0. (m-1)(m-9) = 0. So m = 1 or m = 9.

AI explanation

For the quadratic equation x^2 - (m - 3)x + m = 0 to have equal roots, its discriminant must be exactly zero. The discriminant is calculated as the square of the coefficient of x minus four times the product of the coefficient of x^2 and the constant term, giving (m - 3)^2 - 4(1)(m) = 0. Expanding and simplifying this equation yields m^2 - 10m + 9 = 0, which factors into (m - 1)(m - 9) = 0. Solving this gives the values of m as 1 and 9, resulting in the set {1, 9}.

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