Question

15 points plz help

Let k be constant. If the equation (x+2)(x+3) = kx +2 has equal positive roots,then x=?

Answer 1

Expand the brackets: x²+5x+6=kx+2. So x²+(5-k)x=-4.
To complete the square on the left we add ((5-k)/2)² to each side:
(x+(5-k)/2)²=((5-k)/2)²-4.
Take a look at the right hand side: when this is zero we have equal roots:
((5-k)/2)²=4, (5-k)/2=±2, 5-k=±4, k=5∓4, giving us k=1 or 9.
(5-k)/2 is 2 or -2. So if k=9, (x-2)²=0. And x=2 is the positive root.