<s>[INST] Find the possible values of $k$, if the equation has equal roots. The equation $x^2+5k=kx+x+19$ has equal roots. Find the possible values of $k$.
Um having problem in rearranging the equation;
$x^2+5k-kx-x-19=0$
$x^2+k(5-x)-x-19=0$
What is the next step?
 [/INST] Try this
$$ x^2 - (k +1)x + (5k - 19) = (x - a)^2 = x^2 -2a + a^2 = 0$$
So now by inspection , $$5k - 19 = a^2 $$  $$(k+1) = 2a$$
$$ k = 2a -1 $$
$$5(2a -1) - 19 = a^2$$
$$a^2 - 10a + 24 = 0$$
$$(a -6)(a-4) = 0$$
$$a = 6 \ , \ a = 4$$
$$k = 2(6)-1 \ , \ k = 2(4)-1 $$
$$k= 11 \ , \ k = 7 $$
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