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Find the largest integer $k$ such that the equation
5x^2 - kx + 8 - 20x^2 + 45 = 0
has no real solutions.

 Jun 27, 2024
 #1
avatar+1252 
+1

First, let's simplify this equation. Combining all like terms, we have

\(-15x^2-kx+53=0\)

 

Now, in order for the value of x to not be real, then the descriminant must be negative. 

Thus, we have

\({\left(-k\right)^2-4\left(-15\right)\cdot \:53} < 0\)

 

Now, we simplfy solve for k. We get that

\(k^{2}-3180 < 0\\ k^2 < 3180\\ -2\sqrt{795} <2\sqrt{795}\\ -56.39 < k < 56.39\)

 

The largest integer k can be that fits in the range given for k is 56. 

 

So our answer is just 56. 

 

Thanks! :)

 Jun 27, 2024

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