What is the value of k so that y = x^2 + 3k + 4 and x + y + 3 = 0 meet at only one point?

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There is an error in your question, the equation of the curve should be given as y = x^2 + 3k + 4

We have to determine the value of k which ensures that the curve y = x^2 + 3k + 4 and the line x + y + 3 = 0 meet at only one point.

x + y + 3 = 0

=> y = -x - 3

substitute in y = x^2 + 3k + 4

=> -x - 3 = x^2 + 3k + 4

=> x^2 + x + 3k + 7 = 0

This equation should have only one solution. That is possible if

(1)^2 - 4*1*(3k + 7) = 0

=> 1 - 12k - 28 = 0

=> 12k = -27

=> k = -27/12

**The required value of k is -27/12**

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