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

We need to find when the equation $f_1(x) = f_2(x)$ has two distinct real roots.

This means we're looking for when:

$x^2 + 11x + n = x$

has exactly two different real solutions.

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Let's rearrange the equation by moving everything to one side:

$x^2 + 11x + n = x$

$x^2 + 11x + n - x = 0$

$x^2 + 10x + n = 0$

We now have a quadratic equation in standard form $ax^2 + bx + c = 0$ where:

$a = 1$

$b = 10$

$c = n$

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For a quadratic equation $ax^2 + bx + c = 0$ to have two distinct real roots, we use the discriminant.

The discriminant is $\Delta = b^2 - 4ac$

If $\Delta > 0$: Two distinct real roots

If $\Delta = 0$: One repeated root (not what we want)

If $\Delta < 0$: No real roots (complex roots)

Since we want two distinct real roots, we need:

$\Delta > 0$

$b^2 - 4ac > 0$

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Substituting our values $a = 1$, $b = 10$, and $c = n$:

$10^2 - 4(1)(n) > 0$

$100 - 4n > 0$

$100 > 4n$

$\frac{100}{4} > n$

$25 > n$

Therefore: $n < 25$

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We need:

$n$ to be a positive integer

$n < 25$

The largest positive integer that satisfies $n < 25$ is $n = 24$.

When $n = 24$:

$\Delta = 100 - 4(24) = 100 - 96 = 4 > 0$

When $n = 25$:

$\Delta = 100 - 4(25) = 100 - 100 = 0$

This gives one repeated root, not two distinct roots.

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The largest positive integer $n$ for which the equation has two distinct real roots is $n = 24$.

The discriminant $b^2 - 4ac > 0$ is the key tool for ensuring a quadratic has two distinct real roots.