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Quadratic Equations

If f 1 x = x ...

Question

If $f_{1} (x) = x^{2} + 11 x + n$ and $f_{2} (x) = x$ , then the largest positive integer n for which the equation $f_{1} (x) = f_{2} (x)$ has two distinct real roots, is___

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Solution

$f_{1} (x) = x^{2} + 11 x + n, f_{2} (x) = x f_{1} (x) = f_{2} (x) \Rightarrow x^{2} + 11 x + n = x x^{2} + 10 x + n = 0$
For this equation to have two distinct real roots, D > 0
$100 - 4 n > 0 \Rightarrow n < 25$
Hence, the largest value of n is n = 24.

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