If the quadratic equation- ax2-2cx-b-0 ax2-2bx-5x-c-0 b - c hava a common root then find the value of a-4b-4c-

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Standard XII

Mathematics

Geometric Progression

If the quadra...

Question

If the quadratic equation, $a x^{2} + 2 c x + b = 0$ & $a x^{2} + 2 b x - 5 x + c = 0 (b \neq c)$ hava a common root then find the value of $a + 4 b + 4 c$ .

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Solution

Considering the equations to be:

$a x^{2} + 2 c x + b = 0$ ...(1)

$a x^{2} + 2 b x + c = 0$ ...(2)

(2)-(1) gives

$2 (b - c) x + c - b = 0$

$2 (b - c) x = b - c$

$2 x = 1$

$∴ x = \frac{1}{2}$

From (1)

$a x^{2} + 2 c x + b = 0$

$a {(\frac{1}{2})}^{2} + 2 c \frac{1}{2} + b = 0$

$a + 4 c + 4 b = 0$

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