a+24((a−3)/(b^2 +a))=2a−b^(2 ) a×b^2 =20 would you solve this?


Question and Answers Forum

All Questions Topic List
Others Questions
Previous in All Question Next in All Question
Previous in Others Next in Others
Question Number 5159 by 1771727373 last updated on 24/Apr/16

$a + 24 \frac{a - 3}{b^{2} + a} = 2 a - b^{2}$ $a \times b^{2} = 20$ $w o u l d y o u s o l v e t h i s ?$
	Answered by FilupSmith last updated on 24/Apr/16

	${a b}^{2} = 20 \Rightarrow b^{2} = 20 a$ $∴ a + 24 \frac{a - 3}{b^{2} + a} = 2 a - b^{2} \Rightarrow a + 24 \frac{a - 3}{\frac{20}{a} + a} = 2 a - \frac{20}{a}$ $a + 24 \frac{a - 3}{\frac{20 + a^{2}}{a}} = \frac{2 a^{2} - 20}{a}$ $a + 24 \frac{a^{2} - 3 a}{20 + a^{2}} = \frac{2 a^{2} - 20}{a}$ $a^{2} + 24 a (a^{2} - 3 a) = (2 a^{2} - 20) (20 + a^{2})$ $a^{2} + 24 a^{3} - 72 a = 40 a^{2} + 2 a^{4} - 20^{2} - 20 a^{2}$ $- 19 a^{2} + 24 a^{3} - 2 a^{4} - 72 a + 400 = 0$ $19 a^{2} - 24 a^{3} + 2 a^{4} + 72 a - 400 = 0$ $2 a^{4} - 24 a^{3} + 19 a^{2} + 72 a - 400 = 0$ $c o n t i n u e$
	Answered by Yozzii last updated on 28/Apr/16

	$a (b^{2} + a) + 24 (a - 3) = (2 a - b^{2}) (b^{2} + a)$ ${a b}^{2} + a^{2} + 24 a - 72 = 2 {a b}^{2} + 2 a^{2} - b^{4} - {a b}^{2}$ $b^{4} - a^{2} + 24 a - 72 = 0$ $b^{4} = a^{2} - 24 a + 72$ $b^{4} = \frac{400}{a^{2}}$ $∴ 400 = a^{4} - 24 a^{3} + 72 a^{2}$ $a^{4} - 24 a^{3} + 72 a^{2} - 400 = 0 ()$ $L e t f (x) = x^{4} - 24 x^{3} + 72 x^{2} - 400 .$ $B y t r i a l a n d e r r o r, a n d a p p l i c a t i o n$ $o f t h e i n t e r m e d i a t e v a l u e t h e o r e m,$ $a r o o t f o r t h e e q u a t i o n e x i s t s i n$ $t h e i n t e r v a l [20.5, 20.6] .$ $f^{^{'}} (x) = 4 x^{3} - 72 x^{2} + 144 x$ $B y t h e N e w t o n - R a p h s o n m e t h o d$ $t h e n + 1 t h a p p r o x i m a t i o n t o t h e$ $r o o t o f f (x) = 0, g i v e n t h e n t h a p p r o x i m a t i o n,$ $i s f o u n d b y$ $x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{^{'}} (x_{n})}$ $x_{n + 1} = \frac{3 x_{n}^{4} - 48 x_{n}^{3} + 72 x_{n}^{2} + 400}{4 x_{n}^{3} - 72 x_{n}^{2} + 144 x_{n}}$ $L e t x_{1} = \frac{20.5 + 20.6}{2} = 20.55 .$ $\Rightarrow x_{2} \approx 20.540974$ $\Rightarrow x_{3} \approx 20.540961$ $\Rightarrow x_{4} \approx 20.540961$ $∴ a \approx 20.541 . N o t e t h a t () h a s m o r e$ $t h a n j u s t o n e r e a l r o o t . T h i s s a m e$ $m e t h o d c o u l d b e u s e d t o s e a r c h f o r$ $o t h e r r e a l r o o t s . S i n c e b^{2} = \frac{20}{a}, i f$ $b \in R \Rightarrow a > 0 ∴ b = \pm \sqrt{\frac{20}{20.541}} \approx \pm 0.9867$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum