চলক (গণিত)

৩/বহুঘাত বিশিষ্ট চলক

এক চলক সংবলিত দ্বিঘাত সমীকরণের আদর্শ রূপ ${\displaystyle ax^{2}+bx+c=0}$ । এখানে ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ বাস্তব সংখ্যা এবং ${\displaystyle {\displaystyle a\neq 0}}$ ।[2]

	যথার্থতা
${\displaystyle ax^{2}+bx+c=0}$
⇒ ${\displaystyle (ax^{2}+bx+c)\times a=0\times a}$	উভয় পক্ষকে ${\displaystyle a}$ দ্বারা গুণ করে
⇒ ${\displaystyle a^{2}x^{2}+abx+ac=0}$
⇒ ${\displaystyle (ax)^{2}+2}$ • ${\displaystyle (ax)}$ • ${\displaystyle {\frac {b}{2}}+\left({\frac {b}{2}}\right)^{2}-\left({\frac {b}{2}}\right)^{2}+ac=0}$
⇒ ${\displaystyle (ax)^{2}+2}$ • ${\displaystyle (ax)}$ • ${\displaystyle {\frac {b}{2}}+\left({\frac {b}{2}}\right)^{2}=\left({\frac {b}{2}}\right)^{2}-ac}$
⇒ ${\displaystyle (ax)^{2}+2}$ • ${\displaystyle (ax)}$ • ${\displaystyle {\frac {b}{2}}+\left({\frac {b}{2}}\right)^{2}=\left({\frac {b^{2}}{2^{2}}}\right)-ac}$
⇒ ${\displaystyle (ax)^{2}+2}$ • ${\displaystyle (ax)}$ • ${\displaystyle {\frac {b}{2}}+\left({\frac {b}{2}}\right)^{2}={\frac {b^{2}}{4}}-ac}$
⇒ ${\displaystyle \left(ax+{\frac {b}{2}}\right)^{2}={\frac {b^{2}}{4}}-ac}$	${\displaystyle \because a^{2}+2ab+b^{2}=(a+b)^{2}}$
⇒ ${\displaystyle \left(ax+{\frac {b}{2}}\right)^{2}={\frac {b^{2}-4ac}{4}}}$
⇒ ${\displaystyle {\sqrt {\left(ax+{\frac {b}{2}}\right)^{2}}}={\sqrt {\frac {b^{2}-4ac}{4}}}}$	উভয় পক্ষের বর্গমূল করে
⇒ ${\displaystyle ax+{\frac {b}{2}}={\sqrt {\frac {b^{2}-4ac}{4}}}}$
⇒ ${\displaystyle ax+{\frac {b}{2}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2}}}$	${\displaystyle \because ax+{\frac {b}{2}}={\frac {\sqrt {b^{2}-4ac}}{\sqrt {4}}}}$
⇒ ${\displaystyle ax=-{\frac {b}{2}}\pm {\frac {\sqrt {b^{2}-4ac}}{2}}}$	উভয়পক্ষে ${\displaystyle -{\frac {b}{2}}}$ যোগ করে
⇒ ${\displaystyle ax={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2}}}$
${\displaystyle \therefore x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$	উভয় পক্ষকে ${\displaystyle a}$ দ্বারা ভাগ করে