First slide

Theory of equations

Question

The quadratic equations $x^{2} + (a^{2} - 2) x - 2 a^{2} = 0$ and $x^{2} - 3 x + 2 = 0$ have

Moderate

A

no common root for all $a \in R$
B

exactly one common root for all $a \in R$
C

two common roots for some $a \in R$
D

none of these

Solution

Let $a$ be a common root of the equations
$x^{2} + (a^{2} - 2) x - 2 a^{2} = 0$ and $x^{2} - 3 x + 2 = 0$
Then,
$α^{2} + (a^{2} - 2) α - 2 a^{2} = 0$ and $α^{2} - 3 α + 2 = 0$
Now,
$α^{2} - 3 α + 2 = 0 \Rightarrow α = 1, 2$
Putting $α = 1$ in $α^{2} + (a^{2} - 2) α - 2 a^{2} = 0$ , we get
$\Rightarrow a^{2} + 1 = 0$ , which is not possible for any $a \in R$ .
Putting $α = 2$ in $α^{2} + (a^{2} - 2) α - 2 a^{2} = 0$ , we get
$4 + 2 (a^{2} - 2) - 2 a^{2} = 0,$ which is true for all $a \in R$ .
Thus, the two equations have exactly one common root for all $a \in R$ .

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