Question

Find the value (s) of k so that the equation $x^2-11x+k=0$ and $x^2-14x+2k=0$ may have a common root.

Answer 1

Given,

$x^2-11x+k=0$...(1)

$x^2-14x+2k=0$...(2)

$\left(1\right)\times 2=2x^2-22x+2k=0$...(3)

(3)-(2) gives

$x^2-8x=0$

$x(x-8)=0$

$x=0,8$

From (1)

when $x=0\Rightarrow 0^2-11\left(0\right)+k=0\Rightarrow k=0$

when $x=8\Rightarrow 8^2-11\left(8\right)+k=0\Rightarrow k=24$

$(x,k)=\left\{\right(0,0),(8,24\left)\right\}$