First slide

Theory of equations

Question

The number of values of the pair (a, b ) for which the equation $α (x + 1)^{2} + β (x^{2} - 3 x - 2) + x + 1 = 0, \forall x \in R$ is identity

Moderate

A

1
B

0
C

infinite
D

None of these

Solution

The equation $α (x + 1)^{2} + β (x^{2} - 3 x - 2) + x + 1 = 0, \forall x \in R$
is an identify in x if
$\begin{array}{r} α + β = 0 - - - (i) \\ 2 α - 3 β + 1 = 0 - - - - (ii) \\ α - 2 β + 1 = 0 - - - (iii) \end{array}$
Solving Eq. (1) and (2), we get $α = - \frac{1}{5} and β = \frac{1}{5}$
But these values do not satisfy Eq. (3). Hence the given equation cannot be an identify for any values of $α and β$

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