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Find the largest integer  such that the equation

5x^2 - mx + 8 = 4x^2 - 12

has no real solutions.

 Jul 2, 2022
 #1
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Simplify to get:

\(x^2-mx+20=0\)

Now, for this Quadratic to have no real solutions, it should not intersect the x-axis. 

So, the discriminant , which is: \(b^2-4ac\) must be less than zero.

So:
\(b^2-4ac<0\)

\(\iff m^2-80 <0\)  (Now we have to solve this Quadratic inequality for m).

The critical values:  \(m=\pm 4\sqrt{5}\)

Next, sketch (*) and place the critical points, you want the region below the x-axis (Hence, between the critical values):
Thus, \(-4\sqrt{5}

Since hte question is asking about the largest integer m, let us convert this to decimal:

\(-8.94427..

Therefore, the largest integer m, is: \(m=8\) is the desired solution.

 

Hope this helps!

 Jul 3, 2022

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