Determine k such that the quadratic equation $x^2 + 7(3 + 2k)+(1 + 3k) = 0$ has equal roots

Assuming the given equation is meant to be x squared plus 7 times the quantity 3 plus 2k times x plus the quantity 1 plus 3k equals 0, we apply the condition for equal roots by setting the discriminant equal to zero. This gives 7 times the quantity 3 plus 2k all squared minus 4 times 1 times the quantity 1 plus 3k equals 0, which expands to 49 plus 84k plus 196k squared minus 4 minus 12k equals 0. Simplifying results in the quadratic equation 196k squared plus 72k plus 45 equals 0, whose discriminant evaluates to a negative number, meaning there are no real values of k that make the roots equal. The correct option is None of these.

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