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For what real values of k does the quadratic 12x^2 + kx + 27 = 8x^2 + 18 + 14x^2 + 32 have nonreal roots? Enter your answer as an interval.

 Aug 9, 2024
 #1
avatar+1897 
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First, let's move all terms to one side and set up a quadratic equation. We have

\(-10x^2+kx-23=0\)

 

Now, in order for the roots to be nonreal, the descriminant must be less than 0. Thus, we can write the equation

\(k^2-920 < 0\)

 

Now, we simply solve for k. We have

\(k^2<920\\ \)

Since we must square k, there are two restrictive intervals for k. We have

\(-2\sqrt{230} <2\sqrt{230}\)

 

So our final answer for k is

\(-2\sqrt{230} <2\sqrt{230}\)

 

Thanks! :)

 Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024

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