Question

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

Comments

The actual answer is -6..but how??

Answer 1

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

Comments

The actual answer is -6 ...but how??

Answer 2

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