10. The range of values of k for which the equation below has two different/distinct real roots is:
x2 - 3kx - k = 0
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Answers
Step-by-step explanation:
You are almost there. You get, correctly, that there are two distinct, real roots if and only if k2−24>0. So this is true if and only if k2>24. Since both sides are greater than 0, we may apply the (positive) square root function to both sides to obtain the final inequality
|k|>24−−√=26–√
recalling that x2−−√=|x| (x∈R). We can then write this as
k>26–√ or k<−26–√.
Observe that these are the two critical points that you found. This is where the situation swaps from not having two, distinct real roots to having them. If you just set the inequality to be an equality and solve, then you find the critical points k±, say, then you just need to test the intervals (−∞,k−), (k−,k+) and (k+,∞). Note that it's actually just enough to test one of these intervals, as the adjacent ones must be different to it: in particular, if you test some k∈(k−,k+) in your inequality and find that it does not satisfy, ie you don't have two real, distinct roots, then you know that for k>k+ and k<k− you will. (In our case, k±=±26–√.)