If x = 3 is a solution of the equation (k + 2)x² - kx + 6 = 0 , find the value of k. Hence, find the other root of the equation.

If x = 3 is a solution of the equation (k + 2)x² - kx + 6 = 0 , then x = 3 satisfies the equation.
Putting x = 3 in equation,

$(k + 2)3^2 - 3k + 6 = 0 \\[0.5em] \Rightarrow 9(k + 2) - 3k + 6 = 0 \\[0.5em] \Rightarrow 9k + 18 - 3k + 6 = 0 \\[0.5em] \Rightarrow 6k + 24 = 0 \\[0.5em] \Rightarrow 6k = -24 \\[0.5em] \Rightarrow k = -\dfrac{24}{6} \\[0.5em] k = -4 \\[0.5em]$

Putting value of k in equation in order to find other root

$\Rightarrow (-4 + 2)x^2 - (-4)x + 6 = 0 \\[0.5em] \Rightarrow -2x^2 + 4x + 6 = 0 \\[0.5em] \Rightarrow 2x^2 - 4x - 6 = 0 \text{ (Multiplying equation by -1) }\\[0.5em] \Rightarrow 2x^2 - 6x + 2x - 6 = 0 \\[0.5em] \Rightarrow 2x(x - 3) + 2(x - 3) = 0 \\[0.5em] \Rightarrow (2x + 2)(x - 3) = 0 \text{ (Factorising left side) } \\[0.5em] \Rightarrow 2x + 2 = 0 \text{ or } x - 3 = 0 \text{ (Zero-product rule) } \\[0.5em] \Rightarrow 2x = -2 \text{ or } x = 3 \\[0.5em] x = -1 \text{ or } x = 3 .$

Hence, the values of k is -4 ,and the other root is -1.