Number of values of a for which the equation ( $a^{2}$ - 5a + 4) $x^{2}$ + ( $a^{2}$ - 1) x + ( $a^{2}$ - 8a + 7) = 0 possesses more than two roots, is__.

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Solution

If a quadratic equation has more than two roots, it is an identity.

For the given equation to be an identity, all the three coefficients should be zero.

$a^{2}$ - 5a + 4 = 0, $a^{2}$ - 1 = 0, $a^{2}$ - 8a + 7 = 0

a = 1 or 4, a = 1 or - 1, a = 7 or 1

All the three will be zero when a = 1 or 1 is a common root.

⇒ Only at one value of a the given equation can be considered as an identity.

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Number of values of a for which the equation ( a2 - 5a + 4) x2 + ( a2 - 1) x + ( a2 - 8a + 7) = 0 possesses more than two roots, is__.

Number of values of a for which the equation ( $a^{2}$ - 5a + 4) $x^{2}$ + ( $a^{2}$ - 1) x + ( $a^{2}$ - 8a + 7) = 0 possesses more than two roots, is__.