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Question Number 165226 by Neils last updated on 27/Jan/22

m a k e x t h e s u b j e c t o f t h e f o r m u l a;

a^{x} + b x + c = 0

Answered by mr W last updated on 28/Jan/22

a^{x} = - (b x + c)

{(a^{\frac{1}{b}})}^{b x} = - (b x + c)

{(a^{\frac{1}{b}})}^{b x + c} = - (b x + c) a^{\frac{c}{b}}

e^{(b x + c) \frac{\ln a}{b}} = - (b x + c) a^{\frac{c}{b}}

- (b x + c) \frac{\ln a}{b} e^{- (b x + c) \frac{\ln a}{b}} = \frac{\ln a}{{b a}^{\frac{c}{b}}}

- (b x + c) \frac{\ln a}{b} = W (\frac{\ln a}{{b a}^{\frac{c}{b}}})

\Rightarrow x = - \frac{c}{b} - \frac{1}{\ln a} W (\frac{\ln a}{{b a}^{\frac{c}{b}}})

Commented by Neils last updated on 27/Jan/22

A v e r y n i c e s o l u t i o n b u t {t h e r e}^{'} s a s l i g h t m i s t a k e f r o m t h e f i f t h l i n e

Y o u o m i t t e d a m i n u s s i g n

Commented by mr W last updated on 28/Jan/22

t h a n k s s i r!

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