The equation ${x^2} - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has :
Solution
<p>${x^2} - 4x + [x] + 3 = x[x]$</p>
<p>$\Rightarrow {x^2} - 4x + [x] + 3 - x[x] = 0$</p>
<p>$\Rightarrow (x - 1)(x - 3) - [x](x - 1) = 0$</p>
<p>$\Rightarrow (x - 1)(x - [x] - 3) = 0$</p>
<p>$\therefore$ $x = 1$</p>
<p>or</p>
<p>$x - [x] - 3 = 0$</p>
<p>$\Rightarrow \{ x\} - 3 = 0$ [As $\{ x\} = x - [x]$]</p>
<p>$\Rightarrow \{ x\} = 3$</p>
<p>But we know, $0 < \{ x\} < 1$</p>
<p>$\therefore$ $\{ x\} \ne 3$</p>
<p>$\therefore$ $x$ has only one solution in ($-\infty,\infty$) which is $x = 1$.</p>
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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