What is the sum of all values of k such that the equation 2x^2-kx+8=0 has two distinct integer solutions?
Thank you for your help! :)
two distinct solutions means that the discriminant is positive. So, we need
k^2 - 64 > 0
also, we need
(k±√(k^2-64))/4 to be an integer.
So, k±√(k^2-64) must be a multiple of 4. and k^2-64 must be an integer.
so, what pythagorean triples do you know that have
a^2+8^2 = k^2
6,8,10: 10±6 = 4,16
15,8,17: 17±15 = 2,32
I guess 10 is the only value. 17 has one non-integer solution.
Maybe k can be negative.
Maybe there are some fractional values, but I can't think of any.
the answer is zero yw
To find the sum of all values of k that satisfy the given condition, we need to determine the conditions under which the quadratic equation 2x^2 - kx + 8 = 0 has two distinct integer solutions.
The quadratic equation can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = -k, and c = 8. To have two distinct integer solutions, the discriminant (b^2 - 4ac) must be a perfect square.
Therefore, we have:
(b^2 - 4ac) = k^2 - 4(2)(8) = k^2 - 64
For the discriminant to be a perfect square, k^2 - 64 must be the square of an integer.
Now, we need to find all the values of k for which k^2 - 64 is a perfect square. We know that the difference of two perfect squares is always a multiple of their sum. In this case, (k^2 - 64) is a perfect square if:
k^2 - 64 = a^2
Rearranging the equation, we have:
k^2 - a^2 = 64
This expression, k^2 - a^2, can be factored using the difference of squares formula:
(k - a)(k + a) = 64
To find all the possible values of k, we need to find all the factor pairs of 64 and solve for k in each case:
1. (k - a)(k + a) = 1 * 64
=> (k - a) = 1 and (k + a) = 64
=> k = 32, a = 31
2. (k - a)(k + a) = 2 * 32
=> (k - a) = 2 and (k + a) = 32
=> k = 17, a = 15
3. (k - a)(k + a) = 4 * 16
=> (k - a) = 4 and (k + a) = 16
=> k = 10, a = 6
4. (k - a)(k + a) = 8 * 8
=> (k - a) = 8 and (k + a) = 8
=> k = 8, a = 0
Therefore, the possible values of k that satisfy the given conditions are 32, 17, 10, and 8.
Finally, to find the sum of all these values, we add them up:
32 + 17 + 10 + 8 = 67
Hence, the sum of all values of k satisfying the given condition is 67.