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Graph of Quadratic Expression

If the length...

Question

If the length of $x -$ intercept made by the graph of $f (x) = 4 x^{2} - 10 x + 4$ is $k$ , then the value of $[k]$ is
(where [.] denotes the greatest integer function)

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Solution

Let $f (x) = 4 x^{2} - 10 x + 4 = 0$ has roots as $α, β$
Now,
$4 x^{2} - 10 x + 4 = 0 \Rightarrow 4 x^{2} - 8 x - 2 x + 4 = 0 \Rightarrow (x - 2) (4 x - 2) = 0 \Rightarrow x = \frac{1}{2}, 2$
$x -$ intercept made by the graph is $k$
$k = | α - β | \Rightarrow k = 2 - \frac{1}{2} = \frac{3}{2}$
$∴ [k] = 1$

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