Multiple choice

For what value of 'k' the equation $\displaystyle \left( k+3 \right) { x }^{ 2 }-\left( 5-k \right) x+1=0$ has coincident roots ?

  1. $1, 13$
  2. $1, 12$
  3. $3, 13$

  4. None of these
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A Correct answer
Explanation

For coincident roots, the discriminant D = b^2 - 4ac must be 0. Here, a = k+3, b = -(5-k), c = 1. So, (5-k)^2 - 4(k+3) = 0. 25 - 10k + k^2 - 4k - 12 = 0. k^2 - 14k + 13 = 0. (k-1)(k-13) = 0. k = 1 or 13.

AI explanation

A quadratic equation ax^2 + bx + c = 0 has coincident roots when its discriminant, b^2 - 4ac, equals zero. Here, a = (k + 3), b = -(5 - k) which is (k - 5), and c = 1. Setting the discriminant to zero gives (k - 5)^2 - 4(k + 3)(1) = 0, which simplifies to k^2 - 10k + 25 - 4k - 12 = 0, or k^2 - 14k + 13 = 0. Factoring this quadratic yields (k - 1)(k - 13) = 0, giving the values k = 1 and k = 13.

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