Answer:If the quadratic equation x^2 - x + k = (k + 1)x - 5 has two equal roots, then the discriminant of the quadratic equation is zero.
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.
In this case, the quadratic equation is x^2 - x + k = (k + 1)x - 5, which can be rearranged as x^2 - (k + 2)x + (k - 5) = 0.
Therefore, the discriminant is:
(k + 2)^2 - 4(1)(k - 5) = 0
Simplifying and solving for k, we get:
k^2 + 4k + 4 - 4k + 20 = 0
k^2 + 24 = 0
(k + 2√6i)(k - 2√6i) = 0
where i is the imaginary unit.
Therefore, the possible values of k are -2√6i and 2√6i.
However, since the problem statement does not mention imaginary roots, the possible value of k that satisfies the condition of having two equal roots is k = -2.
If we substitute k = -2 into the quadratic equation, we get:
x^2 - x - 2 = -x - 5
x^2 = 4
x = ±2
Therefore, the two equal roots of the quadratic equation are x = 2 and x = -2, and k = -2 satisfies the condition of having two equal roots.
Explanation: