f(x)=2x2−146x+c has 2 roots that are positive prime numbers. What is c?

To find the value of c, we need to use the information provided that the given quadratic equation has two positive prime roots.

First, let's set up the equation by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Given that the quadratic equation is f(x) = 2x^2 - 146x + c, we can substitute the values into the quadratic formula:

x = (-(-146) ± √((-146)^2 - 4(2)(c))) / (2(2))

Simplifying this equation:

x = (146 ± √(21316 - 8c)) / 4

Since the roots are positive prime numbers, they must be integers greater than 1. Therefore, we can assume that (146 ± √(21316 - 8c)) is an even number.

Let's consider the positive sign in the ± expression:

146 + √(21316 - 8c) = 4k (where k is an integer)

Simplifying:

√(21316 - 8c) = 4k - 146

Squaring both sides:

21316 - 8c = (4k - 146)^2

Expanding the right side:

21316 - 8c = (16k^2 - 584k + 21316)

Simplifying:

-8c = 16k^2 - 584k

Dividing by -8:

c = -2k^2 + 73k

Now, let's consider the negative sign in the ± expression:

146 - √(21316 - 8c) = 4k (where k is an integer)

Following similar steps as above, we would arrive at:

c = -2k^2 - 73k

So, the value of c can be either -2k^2 + 73k or -2k^2 - 73k, where k is an integer.