Factorize x^2 + (1/a) (bx + c)


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Question Number 191436 by MATHEMATICSAM last updated on 24/Apr/23

$Factorize$ $x^{2} + \frac{1}{a} (b x + c)$
	Answered by Matica last updated on 24/Apr/23

	$x^{2} + \frac{b x + c}{a} = x^{2} + \frac{b}{a} x + \frac{c}{a}$ $= x^{2} + 2 (x) (\frac{b}{2 a}) + {(\frac{b}{2 a})}^{2} - {(\frac{b}{2 a})}^{2} + \frac{c}{a}$ $= {(x + \frac{b}{2 a})}^{2} - (\frac{b^{2}}{4 a^{2}} - \frac{4 a c}{4 a^{2}})$ $= {(x + \frac{b}{2 a})}^{2} - (\frac{b^{2} - 4 a c}{4 a^{2}})$ $= {(x + \frac{b}{2 a})}^{2} - {(\frac{\sqrt{b^{2} - 4 a c}}{2 a})}^{2}$ $= (x + \frac{b}{2 a} + \frac{b^{2} - 4 a c}{2 a}) (x + \frac{b}{2 a} - \frac{b^{2} - 4 a c}{2 a})$ $s o$ $x^{2} + \frac{1}{a} (b x + c) = (x + \frac{b + \sqrt{b^{2} - 4 a c}}{2 a}) (x + \frac{b - \sqrt{b^{2} - 4 a c}}{2 a})$
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