Find the largest integer such that the equation
5x^2 - mx + 8 = 4x^2 - 12
has no real solutions.
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!