Final answer:
To find the value of k and the roots of the quadratic equation, we can use the fact that one root is two more than the other. By applying the quadratic formula and considering the given condition, we can determine the value of k and the roots. The value of k is 7, and the roots of the quadratic equation are -1 and 1.
Step-by-step explanation:
To find the value of k and the roots of the quadratic equation, we can use the fact that one root is two more than the other. Let's start by applying the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the given equation, k = a, (k-8) = b, and (1-k) = c. Substituting these values into the quadratic formula, we get:
x = (-(k-8) ± √((k-8)^2 - 4k(1-k))) / (2k)
Now, since we know that one root is two more than the other, we can set up the following equation:
x + 2 = -x
Simplifying this equation, we get:
2x + 2 = 0
Solving this equation, we find one root as x = -1.
Substituting this root into the equation (kx^2+(k-8)x+(1-k)=0), we can solve for k:
(k)(-1)^2 + (k-8)(-1) + (1-k) = 0
Simplifying this equation, we get:
k - k + 8 - 1 + k = 0
Combining like terms, we find:
8 - 1 = -k
Simplifying further, we get:
7 = k
Therefore, the value of k is 7, and the roots of the quadratic equation are x = -1 and x = -1 + 2 = 1.