if-the-roots-of-2x-2-xn-2x-m-is-5-then-find-4n-m-5-

Question Number 188327 by HeferH last updated on 28/Feb/23

i f t h e r o o t s o f 2 x^{2} - x n = 2 x + m i s 5,

t h e n f i n d : 4 n + m - 5

Answered by Rasheed.Sindhi last updated on 28/Feb/23

2 x^{2} - x n = 2 x + m

2 x^{2} - x n - 2 x - m = 0

2 x^{2} - (n + 2) x - m = 0

S a y α, β a r e r o o t s

G i v e n α = β = 5

α + β = \frac{n + 2}{2} = 5 + 5 = 10 \Rightarrow n = 18

α β = - \frac{m}{2} = 5 \times 5 = 25 \Rightarrow m = - 50

4 n + m - 5 = 4 (18) + (- 50) - 5 = 17

Commented by HeferH last updated on 28/Feb/23

N i c e s o l u t i o n! I f o r g o r a b o u t t h o s e p r o p e r t i e s

o f r o o t s

Answered by Rasheed.Sindhi last updated on 28/Feb/23

A n o t h e r w a y \dots

2 x^{2} - x n = 2 x + m

2 x^{2} - x n - 2 x - m = 0

2 x^{2} - (n + 2) x - m = 0 \dots . (i)

T h e e q u a t i o n h a v i n g r o o t s 5, 5 i s

{(x - 5)}^{2} = 0

x^{2} - 10 x + 25 = 0 \dots \dots \dots (i i)

C o m p a r i n g c o e f f i c i e n t s o f (i) & (i i)

\frac{2}{1} = \frac{- (n + 2)}{- 10} = \frac{- m}{25}

n + 2 = 20 \land - m = 50

n = 18 \land m = - 50

▸ 4 n + m - 5 = 4 (18) + (- 50) - 5 = 17

Answered by Rasheed.Sindhi last updated on 28/Feb/23

2 x^{2} - (n + 2) x - m = 0

∙ ∵ 5 i s r o o t

∴ 2 {(5)}^{2} - (n + 2) (5) - m = 0

50 - 5 n - 10 - m = 0

m + 5 n = 40 \dots \dots \dots . (i)

∙ \frac{d}{d x} {(2 x^{2} - (n + 2) x - m)}_{n = 5} = \frac{d}{d x} {(0)}_{x = 5}

{4 x - (n + 2)]}_{x = 5} = 0

\Rightarrow 4 (5) - n - 2 = 0

\Rightarrow n = 18

(i) \Rightarrow m + 5 (18) = 40 \Rightarrow m = - 50

▸ 4 n + m - 5 = 4 (18) + (- 50) - 5 = 17