SOLUTION: Find the number of integers n that satisfy n^2 < 144 + 24n.

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Question 1209288: Find the number of integers n that satisfy n^2 < 144 + 24n.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

For now I'll consider the equation n^2 = 144 + 24n

Let's get everything to one side and then apply the quadratic formula.
n^2 = 144 + 24n
n^2-24n-144 = 0






or

or
The decimal values are approximate.

Draw out a number line.
Mark -4.970563 and 28.970563 on it.

Let's test an integer value to the left of -4.970563
I'll try n = -5
n^2 < 144 + 24n
(-5)^2 < 144 + 24*(-5)
25 < 24
which is false
We eliminate the subset of integers that are -5 or smaller.

Now try an integer value between -4.970563 and 28.970563
I'll pick n = 0
(0)^2 < 144 + 24(0)
0 < 144
which is true
If n is any of the following integers {-4, -3, ..., 27, 28} then the inequality n^2 < 144 + 24n is true.

Lastly we need to test an integer larger than 28.970563
I'll pick n = 29
n^2 < 144 + 24n
(29)^2 < 144 + 24*(29)
841 < 840
which is false
We can eliminate the set of integers that are 29 or larger

--------------------------------------------------------------------------

To summarize we found the integer solution set is {-4, -3, ..., 27, 28}
There are p = 4 items in {-4, -3, -2, -1}
There are q = 28 items in {1,2,3,...,27,28}
So far that's p+q = 4+28 = 32 nonzero integers accounted for.
Then there's 0 itself to bump the final count to 33.

Or you can use the formula last-first+1 to count the number of consecutive integers in a list.
last-first+1 = 28-(-4)+1 = 28+4+1 = 33


Answer: 33
Answer by ikleyn(52835)   (Show Source): You can put this solution on YOUR website!
.
Find the number of integers n that satisfy n^2 < 144 + 24n.
~~~~~~~~~~~~~~~~~~~~~~~~ 

This given inequality

   n^2 < 144 + 24n


is equivalent to

    n^2 - 24n < 144

    n^2 - 24n + 144 < 144 + 144

    (n-12)^2 < 288

    |n-12| < sqrt(288) = 16.97...

    -16.97... < n-12 < 16.97...


Since we look for integer solutions, the last inequality implies


     -16 <= n-12 <= 16.


It has 16 + 1 + 16 = 33 different solutions for integer n-12.


Hence, the original inequality, given in the problem, has 33 different integer solutions.    ANSWER

Solved.



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