If-x-2-ax-18-x-2-7x-2b-x-c-x-5-Find-a-b-c-

Question Number 210934 by hardmath last updated on 22/Aug/24

If \frac{x^{2} + ax - 18}{x^{2} + 7 x + 2 b} = \frac{x - c}{x + 5}

Find a + b + c = ?

Answered by A5T last updated on 22/Aug/24

(x + 5) (x) + 2 (x + 5) + 2 b - 10 \Rightarrow 2 b - 10 = 0 \Rightarrow b = 5

\Rightarrow \frac{x^{2} + a x - 18}{x^{2} + 7 x + 10} = \frac{x^{2} + a x - 18}{(x + 5) (x + 2)} \Rightarrow x + 2 ∣ x^{2} + a x - 18

\Rightarrow {(- 2)}^{2} + a (- 2) - 18 = 0 \Rightarrow a = - 7

\Rightarrow x^{2} + a x - 18 = x^{2} - 7 x - 18 = (x - 9) (x + 2)

\Rightarrow x - c = x - 9 \Rightarrow c = 9 \Rightarrow a + b + c = - 7 + 5 + 9 = 7

Answered by Rasheed.Sindhi last updated on 23/Aug/24

\frac{x^{2} + ax - 18}{x^{2} + 7 x + 2 b} = \frac{x - c}{x + 5}; a + b + c = ?

\frac{x^{2} + ax - 18}{x^{2} + 7 x + 2 b} = \frac{x - c}{x + 5} \cdot \frac{x + d}{x + d}

{\begin{cases} x^{2} + ax - 18 = x^{2} + (d - c) x - cd \\ x^{2} + 7 x + 2 b = x^{2} + (5 + d) x + 5 d \end{cases}

5 + d = 7 \Rightarrow d = 2

{\begin{cases} x^{2} + ax - 18 = x^{2} + (2 - c) x - 2 c \\ x^{2} + 7 x + 2 b = x^{2} + 7 x + 10 \end{cases}

{\begin{cases} 2 c = 18 \Rightarrow c = 9 \\ 2 b = 10 \Rightarrow b = 5 \\ a = 2 - c \Rightarrow a = 2 - 9 = - 7 \end{cases}

a + b + c = - 7 + 5 + 9 = 7

Commented by RojaTaniya last updated on 26/Aug/24

S i r, n i c e s o l u t i o n .

Commented by Rasheed.Sindhi last updated on 27/Aug/24

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Commented by hardmath last updated on 27/Aug/24

thankyou dear professor