We are given the quadratic equation $x^2 + (k-2)x + 10 - k = 0$. We need to find the range of values of $k$ for which the equation has two distinct real roots, expressing the answer in interval notation.

1. Problem Description

We are given the quadratic equation $x^2 + (k-2)x + 10 - k = 0$. We need to find the range of values of $k$ for which the equation has two distinct real roots, expressing the answer in interval notation.

2. Solution Steps

A quadratic equation $ax^2 + bx + c = 0$ has two distinct real roots if and only if its discriminant, $\Delta = b^2 - 4ac$, is strictly greater than zero, i.e., $\Delta > 0$.

In our case, $a = 1$, $b = k - 2$, and $c = 10 - k$. Thus, the discriminant is:

$\Delta = (k - 2)^2 - 4(1)(10 - k)$

We want $\Delta > 0$, so:

$(k - 2)^2 - 4(10 - k) > 0$

$k^2 - 4k + 4 - 40 + 4k > 0$

$k^2 - 36 > 0$

$k^2 > 36$

This inequality holds when $k > 6$ or $k < -6$. In interval notation, this is $(-\infty, -6) \cup (6, \infty)$.

3. Final Answer

The range of values of $k$ for which the equation has two distinct real roots is $(-\infty, -6) \cup (6, \infty)$.