Answer:
The number of different possible values are {-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}.
Step-by-step explanation:
Given : For an integer n, the inequality
has no real solutions in x.
To find : The number of different possible values of n ?
Solution :
The given inequality is
have no solution then the discriminant must be less than zero.
i.e. ![b^2-4ac](https://tex.z-dn.net/?f=b%5E2-4ac%3C0)
Here, a=1, b=n and c=15
![{n}^{2} - 4 \times 1 \times 15 \: < \: 0](https://tex.z-dn.net/?f=%7Bn%7D%5E%7B2%7D%20%20-%204%20%5Ctimes%201%20%5Ctimes%2015%20%20%5C%3A%20%3C%20%20%5C%3A%200)
![n^2](https://tex.z-dn.net/?f=n%5E2%3C60)
![n](https://tex.z-dn.net/?f=n%3C%5Cpm%20%5Csqrt%7B60%7D)
![n](https://tex.z-dn.net/?f=n%3C%5Cpm%207.75)
i.e. ![- 7.75 \: < \: n \: < \: 7.75](https://tex.z-dn.net/?f=-%207.75%20%5C%3A%20%20%3C%20%20%5C%3A%20n%20%5C%3A%20%20%3C%20%20%5C%3A%207.75)
The integer values are {-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}.