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Q.
If the equation $x^{2}+2x-n=0$ has integral roots, then number of values of $n$ between $1$ & $125$ are
Complex Numbers and Quadratic Equations
Solution:
The equation $x^{2}+2x-n=0$ has integral roots , we need all values of $n (1<\,n <\,125)$ for which roots of the equations are integers
Here n represents the product of roots, therefore we need those values of n for which product of roots cannot exceed $125$
Now, from given equation we have
$n=x^{2}+2x=x (x+2)$
$\therefore n =x (x+2)$
$=\{1\times3, 2\times4, 3\times5, 4\times6, 5\times7, 6\times8, 7\times9, 8\times10, 9\times11, 10\times12$, and the product of next pair $11 \times 13 >\, 125\}$
$\therefore $ Number of values of $n$ between $1$ & $125$ are $10$