Find the range of $k$ for which the equation $x^{2} - 4 x + k = 0$ has distinct real roots.

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Solution

The given equation is $x^{2} - 4 x + k = 0$

Here, $a = 1, b = - 4, c = k$

$∴ D = b^{2} - 4 a c = {(- 4)}^{2} - 4 \times 1 \times k = 16 - 4 k$

The given equation will have real and distinct roots, if

$D > 0 \Rightarrow 16 - 4 k > 0 \Rightarrow 4 k < 16 \Rightarrow k < \frac{16}{4} = 4$

Hence, the given equation will have distinct roots, if $k < 4$

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