Show that there is no real number k for which x² + 2(k + 3)x + 9k = 0 has two equal roots.
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Show that there is no real number for which has two equal roots.
Need a reminder? The key ideaShow
The discriminant of is . If there are two distinct real roots, if two equal real roots, , and if no real roots.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The discriminant. .
Step 2. Equal roots need D = 0. That means .
Step 3. Test this equation. Its own discriminant is , which is negative.
Step 4. Conclude. No real solves , so is never zero.
Answer: is always positive, so the roots are never equal.
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