how to show f(x)=x^4 +2x^3 +5x^2 −16x−20 in the form of (x^2 +x+a)^2 −4(x+b)^2 .


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Question Number 162351 by ZiYangLee last updated on 29/Dec/21

$how to show$ $f (x) = x^{4} + 2 x^{3} + 5 x^{2} - 16 x - 20$ $in the form of {(x^{2} + x + a)}^{2} - 4 {(x + b)}^{2} .$
	Answered by Rasheed.Sindhi last updated on 29/Dec/21

	$f (x) = x^{4} + 2 x^{3} + 5 x^{2} - 16 x - 20$ $g (x) = {(x^{2} + x + a)}^{2} - 4 {(x + b)}^{2}$ ${(x^{2} + x + a)}^{2} - 4 {(x + b)}^{2}$ $= x^{4} + x^{2} + a^{2} + 2 x^{3} + 2 a x + 2 {a x}^{2} - 4 x^{2} - 8 b x - b^{2}$ $= x^{4} + 2 x^{3} + 2 {a x}^{2} - 4 x^{2} + x^{2} + a^{2} + 2 a x - 8 b x - 4 b^{2}$ $= x^{4} + 2 x^{3} + (2 a - 3) x^{2} + (2 a - 8 b) x + a^{2} - 4 b^{2}$ $C o m p a r i n g c o e f f i c i e n t s o f f & g$ $2 a - 3 = 5 \land 2 a - 8 b = - 16 \land a^{2} - 4 b^{2} = - 20$ $a = 4 \land b = 3 \land 4^{2} - 4 . 3^{2} = - 20 ✓$ $g (x) = {(x^{2} + x + 4)}^{2} - 4 {(x + 3)}^{2}$
	Commented by mr W last updated on 29/Dec/21

	$n i c e w o r k i n g!$
	Commented by Rasheed.Sindhi last updated on 29/Dec/21

	$T han X for mentioning my mistake S ir!$
	Commented by Rasheed.Sindhi last updated on 29/Dec/21

	$T \underset{sir!}{\overset{L \underset{f}{o} t}{han}} X$
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