7x^5 −4x^4 +9x^3 +12x^2 +5x−9=0 How many roots of this equation are Negative?


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Question Number 13438 by Nayon last updated on 20/May/17

$7 x^{5} - 4 x^{4} + 9 x^{3} + 12 x^{2} + 5 x - 9 = 0$ $H o w m a n y r o o t s o f t h i s e q u a t i o n$ $a r e N e g a t i v e ?$
	Answered by mrW1 last updated on 20/May/17

	$7 x^{5} - 4 x^{4} + 9 x^{3} + 12 x^{2} + 5 x - 9 = 0$ $7 x^{5} - 9 x^{3} + 5 x - 4 x^{4} + 12 x^{2} - 9 = 0$ $x (7 x^{4} - 9 x^{2} + 5) = (4 x^{4} - 12 x^{2} + 9)$ $7 x (x^{4} - \frac{9}{7} x^{2} + \frac{5}{7}) = 4 (x^{4} - 3 x^{2} + \frac{9}{4})$ $7 x [x^{4} - 2 \times \frac{9}{14} x^{2} + {(\frac{9}{14})}^{2} + \frac{5}{7} - {(\frac{9}{14})}^{2}] = 4 [x^{4} - 2 \times \frac{3}{2} x^{2} + {(\frac{3}{2})}^{2}]$ $7 x [{(x^{2} - \frac{9}{14})}^{2} + \frac{59}{196}] = 4 {(x^{2} - \frac{3}{2})}^{2}$ $x = \frac{4 {(x^{2} - \frac{3}{2})}^{2}}{7 [{(x^{2} - \frac{9}{14})}^{2} + \frac{59}{196}]} = \frac{> 0}{> 0} > 0$ $\Rightarrow t h e r e i s n o n e g a t i v e r o o t!$
	Commented by ajfour last updated on 20/May/17

	$a w e s o m e!$
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