Let S be the set of all real roots of the equation,
3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :
Solution
Let 3<sup>x</sup> = t ; t $>$ 0
<br><br>t(t – 1) + 2 = |t – 1| + |t – 2|
<br><br>t<sup>2</sup> – t + 2 = |t – 1| + |t – 2|
<br><br><b>Case-I</b> : t $<$ 1
<br><br>t<sup>2</sup> – t + 2 = 1 – t + 2 – t
<br><br>$\Rightarrow$ t<sup>2</sup> + 2 = 3 – t
<br><br>$\Rightarrow$ t<sup>2</sup> + t – 1 = 0
<br><br>$\Rightarrow$ t = ${{ - 1 \pm \sqrt 5 } \over 2}$
<br><br>$\Rightarrow$ t = ${{\sqrt 5 - 1} \over 2}$ [ As t $>$ 0]
<br><br><b>Case-II</b> : 1 $\le$ t $<$ 2
<br><br>$\Rightarrow$ t<sup>2</sup> – t + 2 = t – 1 + 2 – t
<br><br>$\Rightarrow$ t<sup>2</sup> – t + 1 = 0
<br><br>D $<$ 0 so no real solution.
<br><br><b>Case-III</b> : t $\ge$ 2
<br><br>$\Rightarrow$ t<sup>2</sup> – t + 2 = t – 1 + t – 2
<br><br>$\Rightarrow$ t<sup>2</sup> – 3t - 5 = 0
<br><br>$\Rightarrow$ D $<$ 0 so no real solution.
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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