solve the equation by factoring: 2x³ - 16x² + 14x = 0

Explanation:

Step1: Factor out the greatest common factor

First, factor out (2x) from the equation (2x^{3}-16x^{2}+14x = 0). We get (2x(x^{2}-8x + 7)=0).

Step2: Factor the quadratic expression

Factor the quadratic (x^{2}-8x + 7). We need two numbers that multiply to (7) and add up to (-8). The numbers are (-1) and (-7). So (x^{2}-8x + 7=(x - 1)(x - 7)). The equation becomes (2x(x - 1)(x - 7)=0).

Step3: Set each factor equal to zero

Set (2x=0), then (x = 0). Set (x-1=0), then (x=1). Set (x - 7=0), then (x=7).

Answer:

(x = 0), (x = 1), (x=7)