For $a
eq b$, if the equation $x^{2} + ax + b = 0$ and $x^{2} + bx + a = 0$ have a common root, then the value of $a + b$ equals

Question

For $a 
eq b$, if the equation $x^{2} + ax + b = 0$ and $x^{2} + bx + a = 0$ have a common root, then the value of $a + b$ equals

Dr. Abubacker · Accepted Answer

(a)
Let $\alpha$ be the common root for both the equations $x^{2} + ax + b = 0$ and $x^{2} + bx + a = 0$, then
$\alpha^{2} + a\alpha + b = 0$
And $\alpha^{2} + b\alpha + a = 0$
$\Rightarrow \ \frac{\alpha^{2}}{(a^{2} - b^{2})} = \frac{\alpha}{b - a} = \frac{1}{b - a}$
$	herefore\ \ \ \ \alpha^{2} = - (a + b)$ and $\alpha = 1$
Hence, $a + b = - 1$

Q. For $a \neq b$, if the equation $x^{2} + ax + b = 0$ and $x^{2} + bx + a = 0$ have a common root, then the value of $a + b$ equals

Explanation

Reviewed by Dr. Abubacker

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