First slide

Theory of equations

Question

If the equations $x^{2} - a x + b = 0$ and $x^{2} + b x - a = 0$ have a common root, then

Moderate

A

$a = b$
B

$a + b = - 1$
C

$a + b = 1$
D

$a - b = 1$

Solution

Let $λ$ be the common root of the given equations Then,
$λ^{2} - a λ + b = 0 and λ^{2} + b λ - a = 0$
On subtracting, we get
$λ (a + b) - (a + b) = 0 \Rightarrow a + b = 0 or, λ = 1$
Putting $λ = 1 in λ^{2} - a λ + b = 0, we get$
$1 - a + b = 0 \Rightarrow a - b = 1$

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