Math Question and Answers


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 80543 by Power last updated on 04/Feb/20

Commented by mr W last updated on 04/Feb/20

$1010$
Commented by Power last updated on 04/Feb/20

$solution sir$
Commented by jagoll last updated on 04/Feb/20

$l e t \sqrt{x^{2} - x} = t$ $(t - 2) (t - 4) (t - 6) . . . (t - 2020) = 1$
Commented by jagoll last updated on 04/Feb/20

$\underset{t = 1}{\sum^{1010}} t_{i} = 2 + 4 + 6 + . . . + 2020$
Commented by mr W last updated on 04/Feb/20

$c o r r e c t i s :$ $\underset{t = 1}{\sum^{1010}} t_{i} = - (2 + 4 + 6 + . . . + 2020)$
Commented by jagoll last updated on 04/Feb/20

$o o y e s s i r . t h a n k y o u$
	Answered by mr W last updated on 04/Feb/20

	$l e t \sqrt{x^{2} - x} = 2 z$ $(2 z - 2) (2 z - 4) . . . (2 z - 2020) = 1$ $2^{1010} (z - 1) (z - 2) . . . (z - 1010) = 1$ $(z - 1) (z - 2) . . . (z - 1010) - \frac{1}{2^{1010}} = 0$ $t h i s e q n . h a s 1010 r o o t s f o r z, s a y$ $z_{1}, z_{2}, . . ., z_{1010}$ $\underset{i = 1}{\sum^{1010}} z_{i} = - (1 + 2 + . . . + 1010) = - 510555$ $\underset{i = 1}{\prod^{1010}} z_{i} = 1 \times 2 \times . . . \times 1010 - \frac{1}{2^{1010}} = 1010! - \frac{1}{2^{1010}}$ $\sqrt{x^{2} - x} = 2 z_{i} (i = 1, 2, . . ., 1010)$ $x^{2} - x - 4 z_{i}^{2} = 0$ $f o r e a c h z_{i} t h e r e a r e t w o r o o t s f o r x :$ $x_{i, 1} a n d x_{i, 2} w i t h$ $x_{i, 1} + x_{i, 2} = 1, x_{i, 1} x_{i, 2} = - 4 z_{i}^{2}$ $s u m o f a l l r o o t s f o r x :$ $Σ x = \underset{i = 1}{\sum^{1010}} (x_{i, 1} + x_{i, 2}) = \underset{i = 1}{\sum^{1010}} 1 = 1010$ $p r o d u c t o f a l l r o o t s f o r x :$ $Π x = \underset{i = 1}{\prod^{1010}} (x_{i, 1} x_{i, 2}) = \underset{i = 1}{\prod^{1010}} (- 4 z_{i}^{2})$ $= {(- 4)}^{1010} {[\underset{i = 1}{\prod^{1010}} z_{i}]}^{2}$ $= 2^{2020} {[1010! - \frac{1}{2^{1010}}]}^{2}$ $= {(2^{1010} \times 1010! - 1)}^{2}$ $\Rightarrow s u m o f a l l r o o t s = 1010 \Rightarrow a n s w e r$ $\Rightarrow p r o d u c t o f a l l r o o t s = {(2^{1010} \times 1010! - 1)}^{2}$
	Commented by Power last updated on 04/Feb/20

	$thanks$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum