Find the slope of the line

By Vieta's formulas, the sum of the roots $\displaystyle x_1 + x_2$ of a quadratic equation $\displaystyle ax^2 + bx + c$ is $\displaystyle -\dfrac ba$ .

$\displaystyle \begin{aligned} x(x - k) & = k + 1 \\ x^2 - kx - (k + 1) & = 0 \\ \implies -\dfrac ba & = -\dfrac{- k}{1} = k \end{aligned}$

Any quadratic equations in x with two roots a and b can be expressed as, $(x - a)(x - b) = 0$ . Which on expansion takes the form, $x² - (a + b)x - ab = 0$ Hence, the sum of the roots = -(coefficient of x in the expanded form)

Now, $x(x - k) = k + 1$ => $x² - kx - (k + 1) = 0$

Hence, sum of the roots $= -(-k) = k$

You can actually factorise:

$x^2 - kx - (k+1) = 0$ $\Rightarrow \left(x-(k+1) \right) \left(x+1 \right)= 0$ $\Rightarrow x = k + 1, - 1$

and the sum of roots is just $k + 1 - 1 = \boxed{k}$ .