Solving equations by factoring or by quadratic formula both will find the zeros. When is it better to use one over the other

Factoring a quadratic is not the same as solving it.

Suppose I want to simplify the expression

x2−4x2+x−6.

I factorize the numerator and denominator, to obtain

(x−2)(x+2)(x−3)(x+2)

then I cancel to simplify it to

x−2x−3.

(I could also perform another step here, but that's not what I want to illustrate.)

My point is that factorizing the two quadratics above produced no solution of any of them. The factorization was performed purely to simplify expressions. So clearly factoring a quadratic is not the same as solving it.

To answer your auxiliary question, completing the square is not a method of factorization.

Consider this:

f(x)=x2+x−6

f(x)+6=x2+x

f(x)+6+124=x2+x+124

f(x)+254=(x+12)2

f(x)=(x+12)2−254.

Above, I used the 'completing the square' technique to show that my function f(x) has a minimum value of −254 , which occurs when x=−12 . This technique did not produce a factoring of f(x) at all, so completing the square is not a factorization method.

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