Factorize-x-2-1-a-bx-c-

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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