+2 votes
in Mathematics by (160 points)
edited by

The factors of 2x2-x+p are rational for what value of p?

by (10 points)
The actual answer is -6..but how??

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2 Answers

+2 votes
by (64.2k points)

We have 2x2-x+p = 0

roots are

x= {1+√(1-4x2p)}/4 , {1-√(1-4x2p)}/4

x= {1+√(1-8p)}/4 , {1-√(1-8p)}/4

for factors to be rational p should be 1/8

i.e p=1/8

by (10 points)
The actual answer is -6 ...but how??
+2 votes
by (7.2k points)

As the quadratic equation is 2x2-x+p = 0

For factors to be rational the discriminant of the equation must be a perfect square.

Discriminant = b2 - 4ac = 1 - 4 x 3 x p = 1 - 8p 

If 1 - 8p = (a)2 where a is any integer then the quadratic equation will have linear factor.

Suppose, a = 7
1 - 8p = (7)2
1 - 8p = 49
-8p = 48
p = -6

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