How do you complete the square of 3x squared minus 2x minus 5 equals 0?

Answer 1

Let f = 3x2 - 2x - 5 = 0.

Let's also jolly this up, with a few more letters:

Let, also, g = 3f and z = 3x.

Then, g = 9x2 - 6x - 15

= z2 - 2z - 15

= (z2 - 2z + 1) - 16.

(This completes the square.)

Now, also let

w = z - 1 = 3x - 1.

Then, g = w2 - 16, which easily factorises, as

g = (w - 4)(w + 4)

= (3x - 1 - 4)(3x - 1 + 4);

= (3x - 5)(3x + 3);

and, since g = 3f,

f = ⅓(3x - 5)(3x + 3)

= (3x - 5)(x + 1),

thus giving the correct factors of f = 3x2 - 2x - 5.

The solution follows from this:

f = 0, exactly when either of the two factors given above is zero; that is, when

x = 1⅔ or x = -1.

Completing the square, as you can see, is not always the easiest way to obtain factors; but it has the advantage of working, in every case.