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quadratic

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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+0 Answers

#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