Multiple choice

The values of k for which the roots are real and equal of the following equation $x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0$ are $k = 2, \dfrac{1}{2}$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For real and equal roots, discriminant D = 0. D = [2(5+2k)]^2 - 4(1)(3(7+10k)) = 0. 4(25 + 20k + 4k^2) - 12(7 + 10k) = 0. 100 + 80k + 16k^2 - 84 - 120k = 0. 16k^2 - 40k + 16 = 0. Divide by 8: 2k^2 - 5k + 2 = 0. (2k-1)(k-2) = 0. Roots are k=1/2 and k=2.

AI explanation

For the equation x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0 to have equal roots, the discriminant (D = b^2 - 4ac) must be zero. Substituting a = 1, b = -2(5 + 2k), and c = 3(7 + 10k) gives [-2(5 + 2k)]^2 - 4(1)(21 + 30k) = 0. Expanding this yields 4(25 + 20k + 4k^2) - 84 - 120k = 0, which simplifies to the quadratic equation 16k^2 - 40k + 16 = 0. Dividing by 8 gives 2k^2 - 5k + 2 = 0; factoring this produces (2k - 1)(k - 2) = 0, so the values of k are 2 and 1/2. Therefore, the statement is True.

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