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How many integer values

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rk110 Newbie | Next Rank: 10 Posts Default Avatar
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How many integer values

Post Mon Oct 16, 2017 5:24 am
If x is an integer, then for how many values of p, the value of x2 + px - 24 will be zero?

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Post Thu Nov 09, 2017 6:34 pm
Whoops, should add a note explaining that last assertion. Since x * (x + p) = 24, we also have p = (24 - x)²/x. (It's safe to divide by x since x = 0 is not a solution.)

Any integer p that's a solution to that quadratic is going to itself create a quadratic with two solutions. For instance, say p = 2. Then we'd have

2 = (24 - x)²/x, or

x² + 2x - 24 = 0

which would factor as (x + 6) * (x - 4).

Since every p value is a factor of 24, every quadratic created by a p will have two integer solutions, meaning that each p corresponds to two values of x.

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Post Thu Nov 09, 2017 6:30 pm
I'm going to assume that the problem also requires p to be an integer.

x² + px - 24 = 0

x² + px = 24

x * (x + p) = 24

So x must be a positive or negative factor of 24. That gives us x = -24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, and 24, or sixteen values, each of which has a corresponding integer solution for p. There's some overlap, though! If x = -24, then p = 23, but if x = 1, again p = 23.

Going further, every value of p corresponds to two values of x, so we only have eight integer values of p.

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