\(k\) āĻāϰ āϕ⧋āύ āĻŽāĻžāύ/ āĻŽāĻžāύāϗ⧁āϞāĻŋāϰ āϜāĻ¨ā§āϝ \((3k+1)x^2+2(k+1)x+k=0\) āĻĻā§āĻŦāĻŋāϘāĻžāϤ āϏāĻŽā§€āĻ•āϰāϪ⧇āϰ āĻŦāĻžāĻ¸ā§āϤāĻŦ āĻ“ āϏāĻŽāĻžāύ āĻŦā§€āϜ āĻĨāĻžāĻ•āĻŦ⧇ āĻšāĻŋāϏāĻžāĻŦ āĻ•āϰ
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\((3k+1)x^2+2(k+1)x+k=0\) āϏāĻŽā§€āĻ•āϰāύāϟāĻŋāϕ⧇ \(ax^2+bx+c=0\) āϏāĻŽā§€āĻ•āϰāϪ⧇āϰ āϏāĻžāĻĨ⧇ āϤ⧁āϞāύāĻž āĻ•āϰ⧇ āĻĒāĻžāχ,
\( a=(3k+1),b=2(k+1)\) āĻāĻŦāĻ‚ \(c=k \)

āϝ⧇āĻšā§‡āϤ⧁ āĻŦā§€āϜāĻĻā§āĻŦ⧟ āĻŦāĻžāĻ¸ā§āϤāĻŦ āĻ“ āϏāĻŽāĻžāύ,
∴āύāĻŋāϰ⧂āĻĒāĻ• \(=0\)
āϏ⧁āϤāϰāĻžāĻ‚, \(b^2-4ac=0 \)

āĻ…āĻ°ā§āĻĨāĻžā§Ž, \({2(k+1) }^2-4×(3k+1)×k=0 \)
āĻŦāĻž, \(4(k^2+2k+1)-4(3k^2+k)=0 \)
āĻŦāĻž, \(k^2+2k+1-3k^2-k=0 \)
āĻŦāĻž, \(-2k^2+k+1=0 \)
āĻŦāĻž, \(2k^2-k-1=0 \)
āĻŦāĻž, \(2k^2-(2-1)k-1=0 \)
āĻŦāĻž, \(2k^2-2k+k-1=0 \)
āĻŦāĻž, \(2k(k-1)+1(k-1)=0 \)
āĻŦāĻž, \((k-1)(2k+1)=0 \)

\(∴k=1 \) āĻ…āĻĨāĻŦāĻž \(-\cfrac{1}{2}\)

\(∴k\) āĻāϰ āĻŽāĻžāύ \(1\) āĻ…āĻĨāĻŦāĻž \(-\cfrac{1}{2}\) āĻāϰ āϜāĻ¨ā§āϝ āĻĒā§āϰāĻĻāĻ¤ā§āϤ āĻĻā§āĻŦāĻŋāϘāĻžāϤ āϏāĻŽā§€āĻ•āϰāϪ⧇āϰ āĻŦā§€āϜāĻĻā§āĻŦ⧟ āĻŦāĻžāĻ¸ā§āϤāĻŦ āĻ“ āϏāĻŽāĻžāύ āĻšāĻŦ⧇ āĨ¤

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