Consider the eqn x²+2x-n =0 where n€N and n€[5,100]. Total no. Of different values of 'n' so that the given eqn has integral roots is-
HERE x=-1+root 1+n,-1-root1+n are the roots and when square root of 1+n is a perfect square the root is an integer.in between 5 to 100 there are 8 possibilities;clearly when n=8,15,24,35,48,63,80,100 then n+1 is a perfect square sothat the roots are integers...thank u
100 is not count here becoz square root of n+1 is 101 it is not a perfect square replace 100 with 99 ok.....
To find the total number of different values of 'n' such that the given equation has integral roots, we can use the discriminant of the quadratic equation.
The discriminant is given by the formula: b² - 4ac, where the quadratic equation is in the form ax² + bx + c = 0.
In this case, the equation is x² + 2x - n = 0. Comparing this with the standard form ax² + bx + c = 0, we have a = 1, b = 2, and c = -n.
Now, for the equation to have integral roots, the discriminant must be a perfect square. In other words, b² - 4ac must be a perfect square.
Substituting the values of a, b, and c into the discriminant formula, we get: (2)² - 4(1)(-n) = 4 + 4n.
For the discriminant to be a perfect square, 4 + 4n must be a perfect square.
Let's find the range of values of 'n' (from 5 to 100) for which 4 + 4n is a perfect square.
First, we can simplify the equation: 4 + 4n = 4(1 + n).
Since 4(1 + n) is divisible by 4, in order to be a perfect square, it must also be divisible by its square root, which is 2.
Thus, the possibilities for n are the numbers satisfying (1 + n) = 2k², where k is a positive integer.
Rearranging the equation, we have n = 2k² - 1.
Now, to find the total number of different values of 'n,' we need to determine the number of different values of k that satisfy the equation.
Note that (1 + n) = (2k²) must be in the range from 5 to 101 (exclusive) since n ∈ [5, 100] and k is a positive integer.
So, (2k²) ∈ [6, 102), dividing both sides by 2, we get k² ∈ [3, 51).
Since k is a positive integer, the values of k that satisfy the equation are k = 2, 3, 4, 5, 6, ..., 7 (up to k = 7, as k² ≈ 49, further values would make n exceed 100).
Therefore, the total number of different values of 'n' is 7.
Hence, there are 7 different values of 'n' in the range [5, 100] for which the given quadratic equation x² + 2x - n = 0 has integral roots.