First slide

Theory of equations

Question

$Number of real values of p for which the equations x^{2} - p x + 8 = 0 and x^{2} + p = 0 have two integral solutions is$

Moderate

A

0
B

1
C

2
D

More than 2

Solution

$Let α and β be two integral solutions of x^{2} - p x + 8 = 0 α + β = p and α β = 8$
$Since α and β are integers, hence (α, β) can be (8, 1), (4, 2), (- 8, - 1) and (- 4, - 2) and possible values of p can be 9, 6, - 9, - 6$
$but only for p = - 9, x^{2} + p = 0 will have two integral solutions.$
$Hence only one value of p . i.e. p = - 9 .$

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