if-the-roots-of-the-equation-ax-2-bx-c-0-are-in-the-ratio-3-4-then-show-that-12b-2-49ac-

Question Number 160564 by Avijit007 last updated on 02/Dec/21

i f t h e r o o t s o f t h e e q u a t i o n {ax}^{2} + bx + c = 0

a r e i n t h e r a t i o 3 : 4, t h e n s h o w t h a t

{12 b}^{2} = 49 ac .

Commented by otchereabdullai@gmail.com last updated on 02/Dec/21

nice!

Commented by cortano last updated on 02/Dec/21

{ax}^{2} + bx + c = 0 {\begin{cases} x_{1} \\ x_{2} \end{cases} \to \frac{x_{1}}{x_{2}} = \frac{3}{4}

{\begin{cases} x_{1} = 3 k \\ x_{2} = 4 k \end{cases} \to {\begin{cases} x_{1} + x_{2} = - \frac{b}{a} \Rightarrow k = - \frac{b}{7 a} \\ x_{1} . x_{2} = \frac{c}{a} \Rightarrow c = {12 ak}^{2} \end{cases}

\Rightarrow c = 12 a {(- \frac{b}{7 a})}^{2}

\Rightarrow {49 a}^{2} c = 12 ab \Rightarrow 49 ac = 12 b

Answered by Rasheed.Sindhi last updated on 02/Dec/21

\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} : \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a} = 4 : 3

3 (\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}) = 4 (\frac{- b - \sqrt{b^{2} - 4 a c}}{2 a})

4 (\frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}) - 3 (\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}) = 0

- 4 b - 4 \sqrt{b^{2} - 4 a c} + 3 b - 3 \sqrt{b^{2} - 4 a c}

- b = 7 \sqrt{b^{2} - 4 a c}

b^{2} = 49 b^{2} - 196 a c

48 b^{2} = 196 a c

12 b^{2} = 49 a c

Commented by Rasheed.Sindhi last updated on 02/Dec/21

\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} : \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a} = 3 : 4

4 (\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}) = 3 (\frac{- b - \sqrt{b^{2} - 4 a c}}{2 a})

- 4 b + 4 \sqrt{b^{2} - 4 a c} = - 3 b - 3 \sqrt{b^{2} - 4 a c}

b = 7 \sqrt{b^{2} - 4 a c}

b^{2} = 49 (b^{2} - 4 a c) = 49 b^{2} - 196 a c

48 b^{2} = 196 a c

12 b^{2} = 49 a c

Answered by Rasheed.Sindhi last updated on 02/Dec/21

L e t t h e r o o t s a r e 3 α & 4 α

3 α + 4 α = - \frac{b}{a} \land 3 α . 4 α = \frac{c}{a}

7 α = - \frac{b}{a} \land 12 α^{2} = \frac{c}{a}

\Rightarrow 12 {(- \frac{b}{7 a})}^{2} = \frac{c}{a} \Rightarrow \frac{12 b^{2}}{49 a^{2}} = \frac{c}{a}

\Rightarrow 12 b^{2} = 49 a c

Commented by Avijit007 last updated on 02/Dec/21

t h a n k s .

Answered by Rasheed.Sindhi last updated on 02/Dec/21

R o o t s : 3 k, 4 k

{\begin{cases} a {(3 k)}^{2} + b (3 k) + c = 0 \\ a {(4 k)}^{2} + b (4 k) + c = 0 \end{cases}

{\begin{cases} 9 {a k}^{2} + 3 b k + c = 0 \dots \dots . (i) \\ 16 {a k}^{2} + 4 b k + c = 0 \dots \dots . (i i) \end{cases}

(i i) - (i) : 7 {a k}^{2} + b k = 0

k (7 a k + b) = 0

7 a k + b = 0

k = - \frac{b}{7 a}; k \neq 0

U p d a t e R o o t s : 3 (- \frac{b}{7 a}) & 4 (- \frac{b}{7 a})

P r o d u c t o f r o o t s :

12 {(- \frac{b}{7 a})}^{2} = \frac{c}{a}

12 b^{2} = 49 a c