Question

If equation $ax^2 + 2cx + b = 0$ and $ax^2 + 2bx + c = 0$ have one root in common, then $a + 4b + 4c$ equals

Answer 1

Given: $ax^2 + 2cx + b = 0$ and $ax^2 + 2bx + c = 0$ have a common root.

Let the common root be $\alpha$.

$\Rightarrow a\alpha^2 + 2c\alpha + b = 0$ and $a\alpha^2 +2b\alpha + c = 0$

Subtracting both the equations,
$\alpha (2c - 2b) + b - c = 0$

$\Rightarrow \alpha = \dfrac{1}{2}$

Substituting $\alpha = \dfrac{1}{2}$ in $a\alpha^2 + 2c\alpha + b = 0$, we get

$\dfrac {a}{4} + 2c \times \dfrac{1}{2} + b= 0$

$\Rightarrow a + 4b + 4c = 0$