COMPLETING THE SQUARE (PENYEMPURNAAN KUASA DUA)

I. Basic Ideas

There are two types of factorised quadratic equations, eg:

i)     x2 + 6x + 8 = (x + 2)(x + 4)

ii)    x2 + 6x + 9 = (x + 3)2

 

In type (i), the factors are different. In type (ii), the factors are the same:

     x2 + 6x + 9 = (x + 3)(x + 3)

 

Completing the square (Penyempurnaan kuasa dua) is about writing the equation in a way so that you can “put a square (kuasa-dua) on a bracket” like the (x + 3)2. (That means converting the equation from y = ax2 + bx + c into y = a(x - p)2 + q. Note the + and - signs... DO NOT get them mixed up!)

 

Now, it is possible to do the same with equation (i) up there. You just have to change the 8:

 

     x2 + 6x + 8 = x2 + 6x + 9 – 1

                         = (x + 3)2 – 1

 

As you can see, I made 8 = (9 – 1). You should be able to see why -- Compare all the blue and red sections with equation (ii) above.

 

Similarly,

     x2 + 6x + 12 = x2 + 6x + 9 + 3

                           = (x + 3)2 + 3

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