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Find the largest integer \(m\) such that the equation

\(5x^2 - mx + 8 = 0\)

has no real solutions.

Apr 4, 2020

#1
+188
0

**When the discriminant of an equation is negative, that means it has no real solutions.**

You can solve these types of problems by plugging it (the equation) into the quadratic formula, and then looking at the discriminant. For this specific problem, since it's asking you to find the greatest integer of m so that there are no real solutions, you will have to make m the biggest it can be, but still making the discriminant less than zero.

Example:

Find the largest integer n such that the equation

6x^2 + nx + 3 = 0

has no real solutions.

1. Plug the equation into the discriminant of the quadratic formula (you can also plug it into the formula and then extract the discriminant out, but I'll only plug in the discriminant; you can do either):

n^2 - 4 * 6 * 3 < 0

(I added a '< 0' since the problem wanted me to find the largest integer n that had NO REAL SOLUTIONS.)

2. Simplify what you can, then isolate n:

n^2 < 72

3. Now, it's sort of guess-and-check. Plug in numbers until you find what the question wants you to find; in this case, it wants me to find the GREATEST INTEGER n such that the equation has NO REAL SOLUTIONS. I tried a few numbers, then got 8 as my answer. This is because 8^2 is 64, which is less than 72. If I go up any further, 9^2 is 81, which is greater than 72, so my answer for this example problem is 8.

Good luck, I hope I helped.

Apr 4, 2020
edited by milkcloud  Apr 4, 2020
#2
+37045
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To have no real solutions , the discrimintant of the quadratic equation must be <0

m^2 - 4ac < 0

m^2 - 160 < 0

m^2 < 160

m < 12.64               so   12

Apr 4, 2020
#3
+12530
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m < 4sqrt10

m > -4sqrt10

Apr 4, 2020