Multiple choice

For what value of $k$, will the equation ${ x }^{ 2 }-\left( 3k-1 \right) x+2{ k }^{ 2 }+2k=11$ have equal roots?

  1. $5$
  2. $9$

  3. Both (a) and (b)
  4. $0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac = 0. Equation: x^2 - (3k-1)x + (2k^2 + 2k - 11) = 0. D = (3k-1)^2 - 4(1)(2k^2 + 2k - 11) = 0. 9k^2 - 6k + 1 - 8k^2 - 8k + 44 = 0. k^2 - 14k + 45 = 0. (k-5)(k-9) = 0. So k = 5 or k = 9.

AI explanation

For a quadratic equation ax^2 + bx + c = 0 to have equal roots, its discriminant must be zero, meaning b^2 - 4ac = 0. In the equation x^2 - (3k-1)x + 2k^2 + 2k - 11 = 0, the discriminant is (3k-1)^2 - 4(1)(2k^2 + 2k - 11) = 0. Expanding this gives 9k^2 - 6k + 1 - 8k^2 - 8k + 44 = 0. Simplifying the terms results in the quadratic equation k^2 - 14k + 45 = 0. Factoring this gives (k-5)(k-9) = 0, so the values of k are 5 and 9.

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